European Mathematical Society - 97 Mathematics education
https://euro-math-soc.eu/msc/97-mathematics-education
enWhere do numbers come from
https://euro-math-soc.eu/review/where-do-numbers-come
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>There is a modern trend in calculus courses to start from application examples and practical problem solving, and from there come to some abstraction and theorems. In this book, Körner presents some basic results in a way that is the opposite of this. It is a strict mathematical top-down approach, formulating definitions, properties and theorems with hard proofs starting from a minimal set of axioms. As the title suggests, his guiding idea is to introduce the number systems on a pure axiomatic basis, loosely following the historical evolution. In this sense it is not really a replacement for a calculus course, but rather a complement to it. It does have the structure of a course text, with a sequence of definitions, theorems and proofs that is interrupted with many exercises and challenges for the reader-student.</p>
<p>One would expect that the topics to be covered and the order in which they are introduced are more or less clear. Nevertheless it is somewhat surprising that rational numbers come first in part 1, and natural numbers are a "special case" in part 2, and finally the reals and complex numbers in part 3. The quaternions and polynomials over a field follow as some kind of "encores".</p>
<p>Rationals without integers sounds almost impossible, but here the historical context comes in as a motivation. Originally people counted quantities and that does not really involve numbers in the sense that one added 3 sheep ad 4 sheep to get 7 sheep, but adding 3 sheep to 4 apples was not something to consider. So, to come to abstract counting numbers, the first thing to do in a Greek tradition is to find an axiomatic system for what we call now the positive natural numbers $\mathbb{N}^+$. The Greek had rationals disguised as ratios of lengths or areas but the Indians and Chinese properly considered rationals as numbers. In this book these are introduced as equivalence classes of (numerator, denominator) pairs where numerator and denominator are elements from the previous set with their rules for adding and multiplication. This gives the strictly positive rationals $\mathbb{Q}^+$. Then zero is introduced first as a place holder, and eventually as a number, and this entails the definition of the negative rational numbers and thus besides inverses for multiplication now also inverses for addition will exist. So we now have the structure of $\mathbb{Q}$. This idea of introducing a new mathematical concept as a couple of items from a previous structure with their known composition rules (adding and multiplication) and then identifying them in equivalence classes, is a technique that is used at several places in this book.</p>
<p>In part 2 the natural numbers are derived via the introduction of 1 as the least positive rational whose (reduced) denominator is 1 and then using an induction to generate all the positive integers as part of the rationals, and eventually zero and the negative integers to give $\mathbb{Z}$. Along the way we learn about long division, Bezout's theorem, and prime numbers. Modular arithmetic is introduced via finite fields leading to Fermat's little theorem, coding theory, the Chinese remainder theorem and encryption. The Peano axioms to define the rational numbers $\mathbb{Q}$ as an ordered field is the ultimate conclusion of part 2. It includes some philosophical considerations about the existence of numbers and the idea of an axiomatic approach to mathematics in general. That includes the Russell paradox, Gödel's theorem, and the consistency problem of mathematics.</p>
<p>Historically, mathematics became a profession somewhere in the 17th century. First Körner introduces extensions like $\mathbb{Q}[\sqrt{2}]$ using again the technique of defining addition, multiplication and an order relation on couples of rational numbers. To come to analysis, it requires the fundamental axiom of an intermediate value and hence the existence of a limit for bounded sequences. Equivalence classes of converging sequences are introduced by identifying sequences with the same limit as equivalent. Pointwise operations can be defined for the sequences and the real numbers in the set $\mathbb{R}$ are then identified with these equivalence classes. The complex numbers are then easily introduced as couples of reals (using again the same trick of defining operations for couples of reals) and introducing limits and continuity in $\mathbb{C}$ is relatively easy. This paves the way to define polynomials over a field and to derive the fundamental theorem of algebra. The main reason for considering zeros of polynomials over a field, in particular with integer or rational coefficients, is that this allows to distinguish between the algebraic and the transcendental numbers. Integral domains and quaternions as generalizations of complex numbers are the remaining items with a short discussion.</p>
<p>Clearly this book is probing the fundamentals of mathematical analysis and will be useful as an extra reading for an introductory calculus course. It will certainly satisfy those readers who are looking for abstraction and who want to extract the maximal number of results from the minimal set of axioms. The historical elements on the side are entertaining but not essential. It is not exactly recreational mathematics, but the text is nicely written. It has many explicit proofs and many exercises and invitations to think about a statement. No solutions are provided though. It is an excellent way to get in touch with the foundations of mathematics at a relatively elementary level.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book gives an axiomatic approach at a first year university level to number systems and some elements from calculus. It could be a complement for a freshmen's calculus course.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/tw-k%C3%B6rner" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">T.W. Körner</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-1084-8806-8 (hbk), 978-1-1087-3838-5 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 59.99 (hbk), £ 24.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">268</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come" title="Link to web page">https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97f30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97f40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97f50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F50</a></li></ul></span>Mon, 25 Nov 2019 09:26:01 +0000Adhemar Bultheel49948 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/where-do-numbers-come#commentsGraph Theory and Its Applications (3rd ed.)
https://euro-math-soc.eu/review/graph-theory-and-its-applications-3rd-ed
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This a comprehensive textbook on graph theory is intended as an advanced undergraduate or introductory graduate course. The previous editions of this book had only the first two authors. This edition is a reorganization and makeover of the previous edition with new material added. The style of the previous edition is maintained, meaning that it is a succession of definitions, examples, applications, theorems, proofs, remarks, with little text in between. The many graph drawings and the numerous examples make it easy to understand what the theory is referring to. Many algorithms are presented as high-level pseudo code, but for those students interested in the programming aspects there are extra notes about implementation and several computer programming projects are formulated as exercises. Each chapter is introduced with a brief summary of its objective and contents, and it ends with a glossary giving one-liner descriptions of the new terms that were introduced in that chapter. The preface explains the concept of the book, gives a brief outline of the contents, and some suggestions for the instructor on how to select material from the book for composing a shorter course.</p>
<p>
New to this edition are many "supplementary exercises" (some with hints or solutions at the end) added after each chapter. They complete the many exercises that were included already after the sections of the chapters. Also the algorithmic aspects are more elaborated. The applications of the chapter on colouring and factoring are extended with examples of scheduling problems, map colouring, and problems in computer science, chemistry, circuit theory, etc.</p>
<p>
The compact enumerating style of writing makes it also an excellent reference work. The extensive index, the list of applications, and the glossary of algorithms added at the end are a welcome help for that. The appendices summarizing necessary elements from mathematics (logic and proof techniques, functions, combinatorics, algebraic structures, complexity), and the additional references: books (organized by subject) and papers (organized by chapter), besides the really impressive list of exercises, are of course useful elements helping the students that have to assimilate the material.</p>
<p>
The main contents is organized in 11 chapters. The first two give the basic definitions about concepts, structures, and representations of graphs. Chapter three and four discuss trees and spanning trees. A tree is one of the most important graph structures. They are for example a key-tool in useful applications such as designing different search and coding algorithms. The fifth and sixth chapters introduce connectivity and (optimal) graph traversals. Applications include reliable networks, and routing problems like the Chinese postman and the travelling salesman problem. The Kuratowski theorem is discussed in chapter seven. It characterizes when a graph is planar (no edge crossings). With chapter eight different kinds of graph colouring and graph factorizations are introduced with the applications mentioned above. Students needing operations research or network theory will be most interested in chapters nine and ten, where directed graphs and network flows are discussed. The last chapter is somewhat shorter. It connects the symmetry of the graph with the number of different colourings for that graph.</p>
<p>
Graph theory is applicable in many scientific disciplines, and the book can serve as a basis for any of these. The text is pretty complete in what is discussed but it remains at a basic level. This implies that the math is not so difficult. Of course sets, partitions, and combinatorics are needed but on an algebraic level, permutation groups and morphisms are about the most advanced concepts that are used (for example in the last chapter). The authors chose not to go into problems of spectral graph theory or other more advanced issues.</p>
<p>
The fact that this is the third edition means that the previous editions were already much appreciated, otherwise there would be no need for a third one. With the evolution in the field and extra experience built up by the authors since the last edition (2006), and new ideas brought in by a third author, it is clear that the new edition is an improvement of a textbook that was already a bestseller. In this case it is not only removing typos or clarifying confusing formulations from the previous edition, but it is extended and reorganized. So this is an excellent basic (but more than an introductory) course on graph theory, based on many years of teaching experience. With a volume this size, it is unavoidable that new unintended typos will sneak in. I found some, but there is a website <a href="https://www.graphtheory.com" target="_lank">www.graphtheory.com</a> where more info is available and where suggestions are welcome. The corrections will also be published there. At least that is what is announced in the book, but at the moment of writing this review (January 2019), not many of the links of the website seem to work (e.g. I could not access the errata list for the previous edition) and I doubt that the website is maintained. The last entry dates back to a conference in 2011.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a comprehensive textbook on graph theory. It is edition three which is a reorganization and partial rewrite of the previous edition and new material is added. Many new exercises are added after each chapter. It includes not only the definitions and properties of graphs, but also discusses some of the applications and the computational algorithms.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/onathan-l-gross" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">onathan L. Gross</a></li><li class="vocabulary-links field-item odd"><a href="/author/jay-yellen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jay Yellen</a></li><li class="vocabulary-links field-item even"><a href="/author/mark-anderson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mark Anderson</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/crc-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">crc press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4822-4948-4 (hbk); 978-0-4294-2513-4 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 63,99 (hbk); GBP 35.99 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">577</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/combinatorics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Combinatorics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.crcpress.com/Graph-Theory-and-Its-Applications/Gross-Yellen-Anderson/p/book/9781482249484" title="Link to web page">https://www.crcpress.com/Graph-Theory-and-Its-Applications/Gross-Yellen-Anderson/p/book/9781482249484</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97k10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97K10</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97k30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97K30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05cxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05Cxx</a></li></ul></span>Tue, 05 Feb 2019 08:31:27 +0000Adhemar Bultheel49082 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/graph-theory-and-its-applications-3rd-ed#commentsMathematics Rebooted
https://euro-math-soc.eu/review/mathematics-rebooted
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Lara Alcock has a degree in mathematics, but her research is about mathematical education and mathematical thinking. She has written two books on these subjects already and won several awards for teaching mathematics. The present book is a bit difficult to classify. It is not a book about her research, neither is it a textbook in mathematics. It is also not a popular math book, or is it? That depends on how you define a popular book about mathematics. It is not a collection of puns, puzzles, paradoxes, and all these other topics that are usually found in such books. On the other hand, you could call it a popularizing book about mathematics because it does not require much prior mathematical education to read. In her introduction she describes the readership as those people who have some affection for mathematics, but who lost track at some point in the past. It could for example be a help for teachers or parents who have to help or teach children and who themselves are missing some mathematical way of thinking. Some people, usually at a later stage of life, start to learn a foreign language, or read history books, or biographies, etc. Why should they not learn some mathematics? If they want to pump up their literacy, they could as well improve on numeracy or polish their knowledge of mathematics.</p>
<p>So what does the book contain? It is some kind of a textbook, but it is a freewheeling kind, not restricted by any kind of prescribed rules of what should or should not be included. There are no delimiting containers of algebra, calculus, or geometry. A simple idea brings along another idea, which leads to a different topic etc. Neither is it just entertainment. There are claims, even some sporadic theorems, and there are proofs (sometimes graphical or geometrical to make them more intuitive).</p>
<p>The book has five chapters, which are five threads of ideas. The first chapter is called Multiplying. It starts with the simple idea that by visualizing n rows with m elements makes multiplication easy and immediately proves the commutativity and distributivity. Also identities like $(a+b)^2=a^2+2ab+b^2$ are easily verified geometrically. The area of a triangle and of course also the Pythagoras theorem have geometric proofs as well. But then there is a trail from the Pythagoras theorem to Pythagorean triples and this in turn leads to the last theorem of Fermat. Along the way, suggestions are made for exercises to be elaborated further (Alcock refrains from formulating them as formal exercises, she just suggests to think a bit longer about some problem).</p>
<p>The other chapters are somewhat similar in nature. In the second chapter, entitled Shapes, the starting point are tessellations of the plane. Again by a geometric proof, it follows that the sum of the angles in a triangle is 180°, and by subdividing a polygon in triangles, the formula can be generalized to polygons. But considerations of regular and semi-regular tessellations also lead to symmetries and Penrose tiling. An interesting remark here is that Alcock also stresses the fact that, even with all the formulas included, the mathematics read as sentences. Thus that one is reading mathematics, just like one would be reading some other formula-free text. Another lesson learned is that mathematicians are often interested in generalizing some result, more than just applying it.</p>
<p>Chapter three is called Adding up. The goal is to arrive at infinite sums, but the starting point is adding fractions. But once more there are easy visual ways to show how to add a finite number of integers. This gives classical formulas for the sum of the first n integers, or the odd or the even ones. This is also the place to introduce proofs by induction. Furthermore she tackles convergence and divergence of series, with the geometric and the harmonic series as a particularly interesting cases.</p>
<p>The chapter on Graphs is about plotting functions in a coordinate system. The motivation here is a word formulation of an optimization problem with several (linear) inequality constraints. Plotting the constraints shows easily where the target function will be optimal. Further explorations lead to circles and polar coordinates, but curiously enough the sine and cosine do not appear because only Cartesian equations are used. A glimpse is given at these topics in three dimensions.</p>
<p>The title of the last chapter is Dividing. It starts with explaining our positional number system. This can lead to rules about divisibility like a number is divisible by 3 if and only if the sum of its digits is divisible by 3. When rational numbers are represented as ratios of integers, the prime factorization is needed to simplify them. A bit more advanced is a discussion of rational and irrational numbers and issues of countability.</p>
<p>In some concluding remarks Alcock reflects on what has been learned by reading this book (she has already summarized the main points after each chapter). She also gives suggestions for further reading, many of them are the more mathematical popularizing books, but also some books inspired by research on and experience in mathematical education.</p>
<p>Anybody interested in mathematics (those who want to learn and those who want to teach) will benefit from reading this book, but in my opinion it is particularly of interest for beginning teachers in secondary schools, especially when they were not explicitly trained in teaching mathematics. Parents too who want to follow up their children can be classified in this category. Furthermore there are those who are sincerely interested in jacking up their long forgotten mathematical knowledge.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>With her experience in teaching mathematics and her research in mathematical education, Lara Alcock has composed this book discussing some simple mathematics for the layperson or a fresh university student. As she is not constrained by any curricular prescription, she is freewheeling by association through the mathematical topics. The content pays special attention to those issues that she has experienced as stumbling stones for students.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/lara-alcock" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lara Alcock</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-198-80379-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 19.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">256</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/mathematics-rebooted-9780198803799" title="Link to web page">https://global.oup.com/academic/product/mathematics-rebooted-9780198803799</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A99</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97d70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D70</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Tue, 20 Feb 2018 17:54:03 +0000Adhemar Bultheel48282 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-rebooted#commentsThe Joy of Mathematics
https://euro-math-soc.eu/review/joy-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The authors directly address the secondary school student pointing them to mathematical issues that are not covered by traditional curricula. They are of course addressing students in the USA, but most of what they mention applies to the European system as well. I doubt it that most of these young adults will spontaneously read this book for fun, but there are always exceptions of course. Clearly, through these students, the authors are indirectly reaching the teachers, or it may well be the other way around.</p>
<p>
The subtitle of the book: <em>Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class</em> catch the spirit. What are all these tricks, techniques, and theorems which are not usually covered in a regular curriculum because of a lack of time? The authors have organised them in five chapters collecting many of them around a central theme. The first chapter is called <em>Arithmetic Novelties</em>. I hesitate to call these "novelties", unless they are novelties for the student who may read about them here for he first time. The "novelties" are classic arithmetic tools but that may have been forgotten because many computations are performed on computing machines nowadays and not so much in the heads of students anymore. Examples are shortcuts for divisibility checks, formulas to sum numbers or squares of numbers, the Euclidean algorithm, and fun things to know about numbers like palindromic, triangular or square numbers, perfect numbers and the likes, and more material of that style.</p>
<p>
The second chapter collects some algebraic items. Here are some classics like the irrationality of the square root of 2, why a division by zero allows to prove anything true or false, and there are again useful computational methods: the bisection method to find a zero, the Horner scheme for polynomial evaluation, and problems like solving Diophantine equations, generating Pythagorean triples, Descartes's sign rule for zeros of polynomials, and more.</p>
<p>
The geometry topics of chapter 3 take more pages, but that is mainly because these require many graphical illustrations. As you might expect, we find here several less conventional proofs of the Pythagorean theorem and several of its possible generalisations. Also many theorems involve circles (not surprising since the authors published a year earlier in 2016 <a href="/review/circle-mathematical-exploration-beyond-line" target="_blank">The Circle. A Mathematical Exploration Beyond the Line</a>, a book completely devoted to such circle theorems). But there are many other properties as well that involve triangles, spirals, polygons, Platonic solids and star polyhedra, and much more.</p>
<p>
The chapter on probability is relatively short. Here the surprise effect of unexpected results are a central theme. Benford's law, coinciding birthdays, the Monty Hall problem and the related paradox of Bertrand's box, the false positive paradox, and the poker wild-card paradox. Other topics are surprising properties of Pascal's triangle and random walks.</p>
<p>
The last chapter is a collection of miscellaneous problems. About the origin of some of the familiar mathematical symbols, compound interest and the rule of 72 to double your investment, the Goldbach conjecture, countability and the different levels of infinity, properties and constructions of the parabola, the speed of a bicycle as a function of the sprocket wheel used, and several others.</p>
<p>
Anyone who is a bit familiar with the literature on popular and recreational mathematics will find that most items collected in this book are not really novelties, and as a gem, they are not really neglected, but they certainly are rarely taught in math class. However, if you know some teenager who loves mathematics, then this will be a fantastic gift. All the content is up to the level of her mathematics and it are marvels and gems, which are most probably novelties to her. The good thing is that everything is not just raising wonder and surprise, but it is explained why it works and proved when appropriate. It is not a regular textbook tough with formal theorems, proofs and exercises. It is kept at an entertaining level. If, as a teacher, you have some spare time within the strict framework of the curriculum, you can use the book as an inspiration for examples that are stimulating the interest of your pupils.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of popular mathematical topics that are brought at the level of secondary school students but that is usually not included in the regular curriculum because of time constraints. Things are explained and proved at an appropriate level, but it is recreational in the sense that it is not a textbook with formal theorems, proofs and exercises.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-geretschl%C3%A4ger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Geretschläger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633882973 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 18.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">300</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.edelweiss.plus/#sku=1633882977&amp;amp;page=1" title="Link to web page">https://www.edelweiss.plus/#sku=1633882977&page=1</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Mon, 08 Jan 2018 20:58:08 +0000Adhemar Bultheel48155 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/joy-mathematics#commentsEssentials of Mathematical Thinking
https://euro-math-soc.eu/review/essentials-mathematical-thinking
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
With a title like "Essentials of Mathematical Thinking" one might expect a philosophical treatise, or possibly a research exposition about cognitive processes and math education. But at the top of the cover, you can see that it is announced as a "Textbook in Mathematics". Since that is what it is: a textbook in mathematics, but a rather unconventional one. Several writers of popular science or recreational mathematics have written books in which they collect mathematical topics that are accessible for a general public and that should illustrate that mathematics can be fun and that there are many practical applications in everyday life involving mathematics. The items discussed in these books can involve integers, prime numbers, geometry, probability, counting problems, logic and paradoxes, games, puzzles, etc. But they are mostly "recreational" or at most they can serve as a source of inspiration for math teachers to embellish their courses and candy-coat the theorems and proofs of the actual textbook.</p>
<p>
Here however, Steven Krantz uses all these entertaining subjects to use them as an actual textbook to teach mathematical awareness and some skills to students who have not the slightest ambition of using mathematics in their further career. For example if undergraduate students are required to broaden their curriculum with some math course. There is no point in imposing mathematical abstraction on them or to force them to memorize proofs of theorems they will never need in life. So the idea is to use all these entertaining subjects to develop their ability to use logic arguments, to solve problems, and to convince them that mathematics is indeed everywhere, but that it is nothing to be afraid of. They will not become better mathematicians in the narrow sense of the word, but at the end of the journey they should have acquired some skills one could call mathematical and they should be more open minded towards mathematics and mathematicians.</p>
<p>
Obviously the book should not have the usual definition-theorem-proof structure and, although formulas have not been abolished completely, there are fewer than in a classical textbook. There are some exercises, but the usual long lists of drilling exercises are absent. Some exercises are meant to drill, some can be more challenging, and chapters are concluded with an open-ended problem. It is like Krantz is telling his story in a stream of consciousness which results in a surprising meandering succession of ideas that will hold the attention of his public or his readers.</p>
<p>
In Chapter 2 the breadth of the field is explored covering many different problems. Some examples: the Monty Hall problem, the four colour problem, minimal surfaces, P vs. NP, Bertrand paradox, etc. This sounds impressive as a starter, but these are actually pretexts for introducing the reader to probability, logarithms,... and to modern tools such as proofs by computer, algebraic computer systems, etc.</p>
<p>
This is followed by seven relatively short chapters. Now problems are solved. Some are classic (When will be the first time after midnight that the hands of an analogue clock will coincide?) others are less classic (Will new years day fall more frequently on a Saturday than on a Sunday? How many trailing zeros will 100! have?...).<br />
To illustrate how ideas are linked, let us consider a section of Chapter 5 as an example. It starts by telling that Kepler derived his laws for the motion of the planets not by solving equations but by analysing observed data (Newton came later), which leads to the meaning of average and standard deviation, which in turn leads to big data and their analysis such as DNA used in forensics, social studies based on Street View and other big sets of data collected by companies such as Google. A remarkable arc that connects Kepler to Google.<br />
Furthermore many of the classics are passing by on the catwalk: the pigeonhole principle, conditional probability, Benford's law, lottery and roulette problems, Conway's Game of Life, Towers of Hanoi, Buffon's needle problem, Euler's characteristic, sphere packing, Platonic solids, voting systems, interpretation of medical tests, facial recognition, wavelets, prisoner's dilemma, Hilbert's hotel and others. Some of these are worked out and actually solved, others are only mentioned as illustration of what is possible, or what they have been used for.</p>
<p>
Up to this point, the text is easily accessible with minimal mathematical background. In the remaining chapters, somewhat more is needed. Chapter 10 is about cryptography (explaining the basics of RSA encryption), the next one gathers some diverse discrete problems (a.o. divergence of the harmonic series, surreal numbers, graphs and the bridges of Königsberg, scheduling problems), and finally a chapter with more advanced problems (Google's Pagerank, needle problem of Kakeya, non-Euclidean geometry, the area of a circle as the limit of the area regular polygons).</p>
<p>
Besides mathematical monographs, Steven Krantz has written books on how to write mathematics, and some books that may be considered as introductions to mathematics for a general public and he won several prizes for his writing. He wrote also one on mathematical education before: <em>How to Teach Mathematics</em> (3rd ed., AMS, 2015) which is about "how" one has to teach. This one is about "what" to teach to a particular type of students. Whether he has experience teaching the "essentials of mathematical thinking" using the material presented in this textbook, I do not know. It might not be a bad idea for students that are somehow obliged to take a math course but that have no the intention to take subsequent courses. I have no information about experiments with this type of course. It would certainly be interesting to know the results.<br />
The text is typeset in LaTeX with the quality of lecture notes. There are many illustrations, but pictures do not have the resolution of high professional quality, and some are not really necessary (a picture of 3 arbitrary dice is not really helpful in solving a probability problem). There are many line drawings too which are usually quite helpful, but by resizing them to fit properly on the page, sometimes circles are distorted and become ellipses or the text in the figure is stretched and resized out of proportion. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of mathematical anecdotes and applications of mathematics that are presented in the form of a textbook. It is the intention that this can be used as a broadening course for (undergraduate) students who do not have an ambition to take further mathematics courses. The topics mainly deal with numbers, geometry, probability, and logic. Analysis is less represented. </p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/steven-g-krantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven G. Krantz</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-and-hallcrc-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chapman and Hall/CRC Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-138-19770-1 (pbk); 978-1-138-04257-5 (hbk); 978-1-315-11682-2 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 44.99 (pbk); £ 115.00 (hbk); £ 40.49 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">336</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701" title="Link to web page">https://www.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Sat, 25 Nov 2017 07:02:54 +0000Adhemar Bultheel48043 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/essentials-mathematical-thinking#commentsHow to Write and Publish a Scientific Paper (8th ed.)
https://euro-math-soc.eu/review/how-write-and-publish-scientific-paper-8th-ed
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book has a long history and was initiated by a seminar given by the second author and a paper he wrote. The response was so positive that it was made into a book that has known several translations and is used as a textbook for courses worldwide. Since the sixth edition Barbara Gastel has joined in and with this new edition, she has become the first author since Robert Day has been emeritus for a while now. The evolution of digital publishing has revolutionized the scientific publishing landscape, which made a new revised edition necessary (the previous one is from 2011). New items are for example the ORCID (that is a unique digital identifier distinguishing an author from any other researcher), the archiving of your (published) paper, warnings against predatory journals, digital poster presentations. There is also a new chapter on editing your own work before publishing, which is somehow a summary of what has already been said in previous chapters.</p>
<p>
Barbara Gastel has a position in biomedical sciences, and Robert Day was teaching English. Their backgrounds clearly show in the focus they have put in the book. Even though there are many general guidelines on how to disseminate your research results, and it certainly introduces inexperienced students to the whole process of publishing a research paper, in my opinion the book is very much oriented towards the habits viable in life sciences and there is an emphasis on writing correct English. I am not familiar with publication culture in social sciences, but as far as mathematics is concerned, one should, after reading this book, consult additionally some books or papers that focus on the peculiarities of publishing mathematics. You may find tens if not hundreds of relevant papers and guidelines available on the Web. Most probably your own institution has one. And then there are of course some of the "classics" listed at the bottom of this review.</p>
<p>
Another issue with this book is that this is mainly written for native English speakers/writers and to some extent even for English speaking American students. This is relevant to understand the humor and word play that is included (there are some old and new cartoons for which language is less relevant). Although there is a section on "writing in English as a foreign language", the emphasis there and in the rest of the book is to use correct English sentences and grammar. In fact a lot of attention goes to grammatical issues and common errors. This of course is important in writing in general, hence also in scientific writing</p>
<p>
Another repeated pattern that is used throughout the book is the IMRAD (Introduction, Methods, Results, and Discussion) structure in all writings or presentation, whether it is a paper, a thesis, a seminar, a report, or whatever. To some extent this also applies to mathematical papers, but not as strictly as it is in life sciences. This format is much more suited for empirical papers, and in some journals publishing experimental results, especially in chemistry or biomedical journals, these five words are actually used as section titles. The results section is the main thing and many of the technicalities or formulas are sometimes banned from the paper and are added as "additional material" provided elsewhere. This is not exactly how a mathematical paper should be composed.</p>
<p>
So in this book, guidelines on how to write your paper are following the IMRAD paradigm. After discussing the title, the list of authors, their addresses and e-mails, the I, M (here referring to Methods and Materials), R, and D sections are discussed separately, and the paper is finished with appropriate acknowledgements and references. The intended readership is obviously the community of students who did bot publish before, so the whole process is explained including the selection of a journal, submitting your paper, the refereeing, and how to react to it, and finally the post-refereeing stage of proofreading and publishing. Clearly, besides all the recommendations given, most journals have specific guidelines for authors that should be consulted. This is repeatedly stressed in the book as well. But the book is covering a very broad publishing culture, by discussing also review papers, and letters to the editors, or writing for a general public, composing a conference abstract or report, or how to prepare a poster or an oral presentation, or write a thesis or a project proposal. Once you became an established author you probably are already familiar with how to write a peer review, but there is still some advice given here. Also how to write a book review, give an interview, or write a book proposal. And for the really ambitious, how to become a science communicator.</p>
<p>
So there are many general guidelines on writing. Certainly the part on writing correct English is extensive but not exactly connected to science writing. There are no particular guidelines for writing mathematical papers. The only place where mathematics is explicitly mentioned is when it is discussed in what order the authors should be listed. It is said that sometimes the order is alphabetical "like for example in mathematics". It is almost standard that mathematical papers are written in LaTeX and somewhat less generally accepted that references are managed with BibTeX. These tools are not even mentioned. The role of arXiv, Zentralblatt, and MathSciNet and the Mathematical Subject Classification (MSC) are not discussed. Neither do they mention the UDC classification. But the book is not only about writing or communication in a strict sense, there is also a discussion about ethics, plagiarism but the pitfalls of self-plagiarism are not highlighted. In this respect, I cannot resist to mention this tendency to multiply the number of papers using a process that is somewhat stimulated by this IMRAD structure: just modify the method or the materials and repeat what has been done already in other papers, and in this way you can produce many carbon copies of just one skeleton paper. Such an objectionable publication policy is less common in mathematics, but it can be a problem for numerical computing papers where a slight variation in the method or the equation to which it is applied can duplicate existing papers. This is of course the consequence of the equally disputable policy of evaluating researchers by counting their papers. In this perspective, it is also remarkable that the book does not discuss impact factors. There is only a distinction between "primary" and "secondary" publications. The impact factors for biomedical journals are so much larger than for mathematical journals that this may be a lesser issue there. Anyway, impact factors reduce mathematics to a negligible section in the science publishing landscape. The authors restrict themselves to give as many generally applicable practical guidelines as possible, but they rightfully avoid points that may raise some controversy since such discussion need not be included in an (under)graduate course. Another recent issue that is not discussed is the data life cycle management (DLM) which should ensure that data, results, software, etc. are still available in the long run. An issue of quickly rising importance in a digital age of fake news.</p>
<p>
The book ends with several appendices. The first appendix is a list of abbreviations for words used in journal titles. "Math." is there, but "Comput." for "computer" or "computational" is not. For the benefit of mathematical students I should mention here the useful AMS <a href="http://www.mathontheweb.org/mathweb/annser_f/annser_frames.html">list of journals and abbreviations</a>. The second appendix is a list of jargon words to be avoided with a preferred alternative. That is certainly useful, also for mathematics. Next are lists of magnitudes (from atto to exa) and of many helpful websites, a glossary of terms used in the book, and an extensive list of references (but the ones below are not in the list), and finally a subject index.</p>
<p>
Conclusion: there are a lot of general guidelines for undergraduates who never published a paper before to learn about the process. Especially the guidelines for using correct English are quite useful. For mathematics one may want to read some extra, more specific, guidelines. Of course, as is also mentioned in this book, much can be learned by consulting (good) examples and by imitation.</p>
<h3>
<strong>Some references relevant for mathematical writing</strong></h3>
<ul>
<li>
N. Steenrod, P. Halmos, M. Schiffer, J. Dieudonné <em>How to Write Mathematics</em> l'Enseignement Mathématique, vol. 16, 1970 and <a href="http://bookstore.ams.org/hwm">AMS booklet</a> 1973<br />
in particular P. Halmos' paper of about 30 pages is still recommended.</li>
<li>
S. Krantz, <a href="https://arxiv.org/abs/1612.04888"><em>A Primer on Mathematical Writing</em></a>. AMS, 1996. A second updated edition was published by the AMS in 2017.</li>
<li>
S. Krantz. <a href="http://bookstore.ams.org/matpub/"><em>Mathematical Publishing, A Guidebook</em></a>. AMS, 2005.</li>
<li>
S. Krantz. <a href="http://www.ams.org/notices/200711/tx071101507p.pdf" target="_blank">How to write your first paper</a>. Notices of the AMS, vol. 54, no. 11, 2007.</li>
<li>
N. Higham. <a href="http://epubs.siam.org/doi/book/10.1137/1.9780898719550"><em>Handbook of Writing for the Mathematical Sciences</em></a>. SIAM, 1998.</li>
<li>
T. Tao. <a href="https://terrytao.wordpress.com/advice-on-writing-papers/" target="_blank">On Writing</a>. blog, retrieved October 10, 2017. </li>
</ul>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a textbook giving practical guidelines for preparing a paper or for general scientific communication. The new edition pays attention to recent issues such as ORCID, archiving, digital presentation, electronic submission, etc. The focus is on communication in biomedical science, and less on mathematical reporting.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/barbara-gastel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Barbara Gastel</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-day" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert A. Day</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781316640432 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">344</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.amazon.co.uk/How-Write-Publish-Scientific-Paper/dp/1316640434" title="Link to web page">https://www.amazon.co.uk/How-Write-Publish-Scientific-Paper/dp/1316640434</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97b20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97B20</a></li></ul></span>Fri, 13 Oct 2017 11:29:07 +0000Adhemar Bultheel47933 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-write-and-publish-scientific-paper-8th-ed#commentsStrategy Games to Enhance Problem-Solving Ability in Mathematics
https://euro-math-soc.eu/review/strategy-games-enhance-problem-solving-ability-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In the introduction the authors explain the goal of their book. When two opponents are playing a board game they need some strategy to win or at least maximize the chances not to loose the game. Mastering such strategies is a skill that can also be used in solving a (mathematical) problem. They even give a table drawing parallels between playing (and winning) the game versus the steps in solving a problem. I believe this is only superficial and much better and deeper analogies do exist. Winning a game is just another problem, where you have to first understand the goal, learn the rules, and develop some strategy. By repeated playing, one learns which strategies work and most of all which strategies work consistently and are not just a lucky hap. The meta question is then whether this strategy can be generalized, to other problems of larger dimension or when the rules are inverted or only slightly changed etc. This is a general approach to solve any problem, whether the problem is mathematical or not.</p>
<p>
Once this has been made clear, the following chapters are basically an enumeration of games, stating the rules and the goal of the game. There is also what the authors call a "sample simulation" in which some moves or situations are simulated, pointing to some problems or possibilities, making suggestions, and sometimes asking questions to think about.</p>
<p>
The different chapters group games that are similar or just form variations of the same game or at least they have similar goals and thus may require similar strategies. We have a chapter on tic-tac-toe-like games, one chapter is called "blocking games" which can come in many different forms (nim is and example), another chapter deals with games where the strategy has to be continuously updated while playing and finally a "miscellaneous" chapter where, among others, several classic western games are found like checkers, dominoes, battleship,...</p>
<p>
So far, only the problems were formulated and the reader, or rather the players, are supposed to go through the steps that I sketched in the first paragraph: finding and analysing a winning strategy. The last chapter provides answers and hints to the problems in the previous chapters. What are possible strategies? For example what is the best first move to open the game? Sometimes a mathematical proof may exist for the claim that the one who starts (and makes no mistakes) will always win. However there are no mathematics involved here, although there could have been. And these mathematics need not always be connected with game theory.</p>
<p>
In an appendix, illustrations of the (empty) game boards are given, although their geometry should be clear already from the previous chapters. Even if these pages are cut from the book, they may be too small to play on. I do not see the advantage of adding these pages to the printed book. Perhaps an electronic version could be printed with some magnification factor.</p>
<p>
In conclusion, this is a nice collection of board games, and when pupils will play such games, they will develop some winning strategies for these games, and these skills will probably help in cultivating certain attitudes and perhaps working schemes to tackle mathematical problems. However it was certainly not the intention of the authors to involve mathematics in this book. I think however that it would not be very difficult to hook up several mathematical problems to these games, somewhat like what Matthew Lane did for video games in his book <a href="/review/power-unlocking-hidden-mathematics-video-games" target="_blank">Power Up</a>. But that would be a completely different book because here the focus is just the games and learning how to win them mostly by playing them, thereby avoiding all the mathematics and abstract game theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The author's leitmotif in this book is that when two players are trying to win a board game, they will develop a winning strategy which is very similar to the strategies and skills needed to solve other, more mathematical, problems. So they describe many of these board games with their rules and urge the reader to play these games several times until they see some strategy emerge. In a final chapter they give some hints about what such strategies may look like for each of these games. There are however no mathematics involved as such.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/stephen-krulik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Krulik</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientfic" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">World Scientfic</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-3146-341 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 20.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">136</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/10187" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/10187</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/91a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91A05</a></li></ul></span>Fri, 23 Jun 2017 11:37:34 +0000Adhemar Bultheel47737 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/strategy-games-enhance-problem-solving-ability-mathematics#commentsProblem-Solving Strategies in Mathematics From Common Approaches to Exemplary Strategies
https://euro-math-soc.eu/review/problem-solving-strategies-mathematics-common-approaches-exemplary-strategies-0
<div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even"> </div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/2017_ProblemSolvingStrategies_0.pdf" type="application/pdf; length=25740">2017_ProblemSolvingStrategies.pdf</a></span></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alfred S Posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/stephen-krulik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Krulik</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific-publishing-co-pte-ltd" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">World Scientific Publishing Co. Pte. Ltd.</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4651-63-9 </div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">188</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span>Wed, 07 Jun 2017 16:48:59 +0000Eugenia47703 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/problem-solving-strategies-mathematics-common-approaches-exemplary-strategies-0#commentsThe Power of Computational Thinking
https://euro-math-soc.eu/review/power-computational-thinking
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Computers play an increasingly important role in mathematics and the converse is also true, old and new branches of mathematics are increasingly important in computer science. Traditionally, languages, science, and mathematics were the corner stones and the main tools in education to develop skills needed to understand the world. Obviously those shape our scientific thinking in a broad sense. The authors believe that more and more also a way of computational thinking is another pillar that should support our educational system. It relies on logic and mathematics and results in an algorithmic approach to problem solving. That involves not only a sequential description of successive steps, but it also includes abstraction and generalization (not solving a particular problem, but solve a whole class of similar problems), it requires checking all the details (no loose ends, and including also all the most unlikely situations), and it needs analysis, and it forces to look for optimal (shortest, fastest, ...) algorithms. Note that all these properties also apply to solving a mathematical problem. And yet there is a difference the authors claim. And I tend to agree, since indeed, I also have known students who were good in mathematics, and yet had difficulties to understand or assimilate an algorithm and others were marvelous computer programmers, but were very reluctant when mathematical theorems and proofs were involved.</p>
<p>
The two authors of this book are mostly involved in education and with this book they want to illustrate what this computational thinking means outside the context of a lesson or a school environment. So they chose the subtitle ``Games, magic and puzzles to help become a computational thinker''. They obviously love magic and tricks and much of what they illustrate relies on these. Part of the text is based on their contributions to the online magazine <em>Computer Science for Fun</em> located at <a href="http://www.cs4fn.org/" target="_blank">www.cs4fn.org</a>.</p>
<p>
They start with Jean-Dominique Bauby, editor in chief of the magazine <em>Elle</em> who, after a stroke, was completely paralyzed and could only blink with an eye. Solve the problem `What is the most efficient way for him to communicate?'. This leads to a binary search algorithm and efficient encoding of an alphabet based on a frequency analysis. Similar cutting of the problem in half at each step is illustrated with a card trick to predict the value of a card from a stack. It turns out that this also applies to mechanically selecting punched cards earmarked with a binary system of slits and holes.</p>
<p>
Cut hive puzzles consist of a hexagonal grid partitioned in subsets each containing a number of neighboring cells. If such a subset had <em>n</em> cells, then the numbers 1,2,...,<em>n</em>need to be placed in these cells, but no cells with the same number can touch each other. This involves not only designing the successive logical steps and rules of the type if... then... for a solution method but it also introduces another important element: patterns. Pattern matching and recognition is important for computational thinking.</p>
<p>
Other puzzles are more like traveling salesman problems: visit a number of squares of a grid with knight moves, or visit a number of interesting tourist attractions in a city, or the classical Königsberg bridge problem. These require a clear representation of the data. In this case clarifying the problem structure via graphs.</p>
<p>
Then a big leap is taken to artificial intelligence. How can a robot learn? Although they don't name it, they touch upon the basics of genetic algorithms, and discuss one of the earliest chatbot ELIZA, and the Turing test. A logical next step is to illustrate the idea of a simple neural network. Since it is a binary example, it also connects with logic binary circuits and simple Boole algebra. Again it is a big leap to then discuss the question if ever artificial intelligent robots or computers can take over our world. The authors reassure their readers that this will certainly never happen because the internet of things consists of things that are too different and moreover they do not have a reason to dominate the human race. So it would only be possible if we let them do it.</p>
<p>
Here we are halfway through the book and here the authors take a step back and discuss some grid games. There is not only the grid of pixels on a screen, but more important is Conway's game of life, or the spit-not-so game. The latter is a game in which two players have to select words from a list until they have 3 words with the same letter in them.</p>
<p>
The remaining chapters involve more tricks and magic using some simple mathematics and most of all pattern matching. For example patterns implied by prime numbers or detecting patterns in an image using simple digital filters, etc. But also the basic principles of CAT scans and MRI are explained. Psychological misleading or make-belief can help in tricks (a variant of the goat-rabbit-cabbage that have to cross the river, or visual illusions, or Weber's law). This is fun, but it seems not to be related to computational thinking, unless one tries to model the human brain and how it experiences its environment.</p>
<p>
The authors conclude by summarizing what they find to be important for computational thinking. It is not only the algorithmic idea, but also modeling the problem and the situation based on scientific arguments, sometimes it requires heuristics, but logic and pattern matching are always important. The representation of the problem can be important to make abstractions and generalizations and it can help to decompose the problem into smaller subproblems. Of course computational problems arise by certain needs people have, so this interaction is also important, understanding what they want and evaluation of the resulting algorithmic product by collecting feedback from users are definitely of practical importance.</p>
<p>
So, I believe it is clear what the authors want to convey to the reader: what is computational thinking and what is covered by that concept. It is certainly not thinking the way a computer thinks but it is linked with how humans function in everyday life, and how this can be transferred to a computational machine. It is however not so clear to me who they want to convey the message to. Are this the children who have to learn these skills? Are it the policy makers who have to define the education programs? Are it the generally interested readers? I believe they address all of them but do not really bring the message to any of them. There is too much magic, tricks, and puzzles to really bring the concepts to children, and they try to cover a really broad spectrum and they touch only superfluously on some topics. There are key words that keep appearing repeatedly typeset in bold in the text like: logic thinking, decomposition, algorithmic thinking, pattern matching, etc. but there is no systematic treatment. After reading the book it gives the impression that magicians must be the best computational thinkers, and you better start practicing card tricks before engaging in programming a computer. But I believe this is not what they had in mind.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The authors explain what they understand by computational thinking. It is much broader than algorithmic thinking and involves logic, analysis, pattern matching, and even understanding the human-computer interaction. They bring their message illustrating it with with low-level examples, often involving puzzles, card tricks and magic.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/paul-curzon" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paul Curzon</a></li><li class="vocabulary-links field-item odd"><a href="/author/peter-w-mcowan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Peter W McOwan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-78634-183-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 48.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">232</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/q0054" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/q0054</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97q99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97Q99</a></li></ul></span>Sun, 19 Feb 2017 09:49:10 +0000Adhemar Bultheel47468 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/power-computational-thinking#commentsThe Circle. A Mathematical Exploration Beyond the Line
https://euro-math-soc.eu/review/circle-mathematical-exploration-beyond-line
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Alfred Posamentier has (co)authored a few dozen books on mathematics and mathematics education. Several of these are intended to popularize math. In 2012 he co-authored with Ingmar Lehmann a book on triangles <em>The Secrets of Triangles: A Mathematical Journey</em>. Triangles are about the simplest mathematical objects interesting enough to prove many theorems about, and they have many applications. Think of the fact that any polygonal area can be triangulated and if the triangular net is made fine enough, they approximate and subdivide any area or surface.</p>
<p>
In this book, the triangles are replaced by circles as the key objects. It contains, among other things about circles, a large variety of classical and less known —but nevertheless quite interesting— theorems that can be formulated concerning properties of circles. These involve often cyclic polygons (i.e., whose vertices lie on a circle) or polygons whose sides are tangent to it. It will not come as a surprise that also here triangles still play a prominent part in this game too.</p>
<p>
In a first chapter some elementary properties and definitions are recalled. It may be a surprise for some readers that the circle is not the only 2D object that has constant breadth. This property it has to share with the Reuleaux triangle, an "inflated" equilateral triangle that is the intersection of three circles with centers at the vertices and radius equal to the side length. A slightly flattened version of it found an ingenious application in the design of the Wankel engine.</p>
<p>
Chapters 2 and 3 discuss a first set of theorems concerning circles. In most cases this starts with a known theorem like for example Ptolemy's (in a cyclic quadrilateral the product of the diagonals equals the sum of the products of the opposite sides). In the case of a rectangle this reduces to Pythagoras' theorem. Generalizations consider n-gons whose vertices are on a circle. This kind of strategy to discuss a theorem is repeated for other theorems: a formulation of the theorem is given, mentioning its origin and sometimes a few sentences about the mathematician that is behind it, then a proof, and by considering special cases, or sometimes generalizations, seemingly unrelated theorems turn out to be included as well. Almost always, there is a further exploration of the problem involved. For example if the theorem claims the collinearity of 3 points, then it may be followed by an analysis of properties about circumferences or areas of triangles and/or circles that were involved in the proof. In this way we are guided along the theorems of Simson, Miquel, Pascal, Brianchon, Ceva, the butterfly theorem, the nine point circle theorem, the six and the seven circles theorems, the Gergonne point theorem, Poncelet's porism, the arbelos, and Ford circles. When a proof is too complicated, it is not included (notes and references are at the end of the book), and sometimes more technical stuff is moved to an appendix, but most proofs are rather easy to follow and are brought in a reader friendly way with many figures that illustrate the successive steps to be followed in the proofs. So proofs are not hard abstract manipulations of formulas, but they rely heavily on the visualization. The text merely explains what the successive steps are, where for example the equal angles are, or the similar or congruent triangles are and why this should be true. It still requires some mental flexibility and an elementary geometric knowledge to follow each step of the proof but it is easy going.</p>
<p>
Circle packing (in a confined space like a rectangle, a circle, or a triangle) is discussed in a short chapter 4. It is explained how it is related to an application in a computer program <em>TreeMaker</em> to design origami patterns.</p>
<p>
The next 4 chapters deal with geometric constructions. Equicircles (the 4 circles that are tangent to all 3 sides of a triangle, 1 inside and 3 outside the triangle) get their own chapter with a computation of their centers and radii.<br />
Chapter 6 is a discussion of the Apollonius problem. That is how to construct a circle that is defined in different ways by points, lines and other circles like containing 3 given points (PPP), or two points and a tangent line (PPL), or a point and 2 tangent lines (PLL), 2 points and a tangent circle (PPC), etc. There are 10 such possibilities. All constructions should be done with straightedge and compass.<br />
The next chapter introduces reflection in a circle. This transforms circles and lines (i.e. circles with infinite radius) into circles and lines. Some of the previous Apollonius problems can be solved in the reflected setting. It can also help in the construction of Steiner and Pappus chains.<br />
Finally chapter 8 is discussing Mascheroni constructions. The ancient Greek tradition allowed to use only straightedge and compass to do all the geometric constructions. It is impossible to do everything with only the straightedge, but one can do without it as was proved by Mohr and independently by Mascheroni. Of course we cannot draw a straight line with a compass, but one can construct any point on a specified line whenever it is needed. Of course the constructions are more involved, but it is shown that with 5 fundamental constructions everything can be done without the straightedge.</p>
<p>
Chapter 9 is again a short interruption from the mathematics since it gives a very brief survey of how the circle was used in arts, in shaping the landscape, and in architecture. This topic could easily be the subject of a whole mathematical picture book on its own, but I do not think the authors have the intention of being complete here. There are just a few examples and it serves to relax a bit from the mathematics in the previous chapters and in the two chapters to follow.</p>
<p>
The remaining two chapters are indeed back into mathematics, but mainly descriptive.<br />
Chapter 10 is for example more recreational: no proofs, but still many graphs. It's all about circles rolling along a line or a circle: the cycloids, hypocycloids, epicycloids, and related curves. Several of these come with a history like the Aristotle wheel paradox and of course the invention of the wheel itself. For the playful aspect, the Spirograph is clearly the instrument of choice. This chapter is not by the authors but it is contributed by Christian Spreitzer.<br />
The last chapter is about spherical geometry. The circle is the only curve that fits also on a sphere. It is well known that the shortest path between two points on a sphere follows a great circle through these points. This explains the route followed during transatlantic flights. There is also the spherical triangle whose angles can sum up to just below 540° and the counterintuitive hight to which a rope around the equator will rise when its length is increased by 1 meter.<br />
An afterword is contributed by Erwin Rauscher who gives a cultural introduction to the circle. The author is different, but it would easily blend in with Chapter 9 on the use of the circle in arts.</p>
<p>
Like several of Posamentier's previous books, this is a book mostly about mathematics, but the "gentle" version, painted on a cultural and historic canvas. The proofs stress the importance of the visual aspect. I am afraid that much of this geometric kind of reasoning has nowadays been largely replaced by algebraic manipulations which, in my opinion, is regrettable. I am old enough to have had this geometric education in secondary school still largely influenced by the Greek tradition of Euclid's Elements, and I remember how I enjoyed solving these geometric "puzzles". It may be one of the reasons that made me decide to become a mathematician. I truly enjoyed re-living this happy experience of my youth by reading this book since during my math studies at the university and in my later career I never used or needed this kind of geometric argumentation.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Like in several of his previous books, Alfred Posamentier, this time in collaboration with Robert Geretschläger, brings this "gentle" kind of mathematics for a broader public. This time it are reflections on geometric problems that are sketched on the canvas of a cultural and historic background. There are proofs but these rely strongly on the many graphics and the geometric constructions. Therefore the mathematics stay away from the dull abstract algebraic formula manipulations that may repulse students.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alfred Posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-geretschl%C3%A4ger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Geretschläger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633881679 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 25.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">349</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.penguinrandomhouse.com/books/539658" title="Link to web page">http://www.penguinrandomhouse.com/books/539658</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97g40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97G40</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/52c26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52C26</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span>Fri, 02 Dec 2016 12:34:45 +0000Adhemar Bultheel47310 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/circle-mathematical-exploration-beyond-line#comments