Recent book reviews

Five recent book reviews

Magnificent Mistakes in Mathematics

Alfred S. Posamentier and Ingmar Lehmann
Prometheus Books
Price (tentative): 
24 USD
Short description: 

It's mistakes galore. Mistakes of all ages and of all sorts and in all areas of mathematics, logic, and statistics. Historical mistakes made by great mathematicians (yes, they are not immune). Mistakes regularly made by students (the typical beginners mistakes much to the exasperation of their hopeless teachers). Mistakes made on purpose for the fun of it (mistakes with a pun). Easily mistaken results because they are counter intuitive (the puzzling ones). You will certainly meet some that you made yourself, but many others that you did not even think of.

URL for publisher, author, or book:
MSC main category: 
00 General
MSC category: 

We all learn from our mistakes (and not only the mathematical ones). This is one of the reasons these authors wrote the book. By showing what kind of mistakes can be made in mathematics, and to what absurd conclusions that may lead, it is hoped that the reader understands better the rules of the game and be more careful in jumping to conclusions.

However, we may learn not only from our own mistakes. There have been many historical mistakes, made by leading mathematicians. The first chapter is a collection of such examples. Pythagoras was mistaken when he thought that nature could be completely explained with natural numbers and their ratios. There have been historical mistakes in the calculation of $\pi$, and many wrong attempts have been made to prove Fermat's last theorem, Goldbach's conjecture, or solve the 4 colour problem, and many other such famous problems. Galileo, Euler, Fermat, Legendre, Poincaré, Einstein, they all made mistakes and often in published papers. Gauss seems to be a glorious exception to this rule. No errors are known in his published papers. This chapter is an enumeration of summaries of these historical errors, although a complete book could be devoted to each of them, why and how the wrong conclusion was made and what kind of research this has started. For example, a wrong calculation of a notorious gambler Chevalier de Mérimé caused him to loose repeatedly. He asked Pascal to explain what seemed to him a paradox, and the correspondence between Pascal and Fermat on this problem can be considered to be the start of probability theory. And we all know that the attempts to prove Fermat's last theorem has given rise to a many new mathematical results.
The subsequent chapters discuss arithmetical, algebraic, geometrical and statistical mistakes. Here we find many obvious errors that are commonly committed by students like division by zero, or violating the rules of distributivity ($\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ or $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$ and the likes). Also jumping too soon to a general conclusion is common. Infinite sums, and working with $\infty$ often leads to false results. Of a somewhat different nature are the rounding errors of digital calculators or computers that may play dirty tricks on us. Sometimes wrong logic may lead to correct answers. Arguments are then needed to convince the student of the low score notwithstanding the fact that the final result was correct. All these are familiar and teachers are desperate when students keep persistently sinning against them. Many examples of mistakes can however be reduced to the same error made under a slightly disguised form. So there is basically a lot of repetition which makes these chapters a bit dull from time to time for readers that are well beyond these rookies mistakes. There are however also errors that are counter intuitive or that have some pitfalls and that are often used in quizzes or to astonish the innocent reader with an apparent paradox. For example suppose the earth is a perfect sphere. Put a rope around the equator and enlarge it by 1 meter. Then keep this longer rope at an equal distance above the surface. Can a mouse pass under the rope? Our intuition says no. However, computation reveals a quite different result. The cat can pursue the mouse at the other hemisphere passing below the rope quite easily. Amusing and puzzling examples are the optical illusions and impossible figures in a geometrical context. A famous geometrical missing square puzzle showing that 65 = 64 is attributed by the authors to Lewis Carroll, although the principle is much older. It has happened that mistakes were deliberately introduced as a prank. Martin Gardner presented in his April 1975 column of Scientific American a map that would require 5 colours, which turned out to be an April fools joke on his readers. Stories like this and the more recreational pitfalls will keep you reading to the end.

All the mistakes are discussed, but sometimes this is not really deep, and sometimes it feels like there is still an untold story behind. The broadness of the examples that are covered (the examples mentioned above are just a tiny sample from a vast set) prevents the inclusion of further details, but an author like Martin Gardner for example could have written a full column including the history, background, generalizations and variations on some of the issues, while here it's more like a sequential enumeration, i.e., a catalog, of the same. All historical facts e.g., are essentially restricted to the first chapeter. In brief: the authors present a not always very deep, but a broad and diverse collection of examples of what people can, and unfortunately often will, do wrong when playing with mathematics. May the reader be wiser after finishing this book.

A. Bultheel
KU Leuven

Mathematics Education in Korea Curricular and Teaching and Learning Practices

Edited by J. Kim, I. Han, M. Park, J. K. Lee.
Word Scientific, Singapore
9789814405850 (print)
Price (tentative): 
112.32 &
Short description: 

This book reports mathematical theory, education research, and practices in Korea

MSC main category: 
97 Mathematics education
MSC category: 

This book reports mathematical theory, education research, and practices in Korea. It has been divided into four main themes groups into three parts: Revision of mathematics curriculum, development of mathematics textbooks for use both now and in the future, teaching and learning practices in some mathematical fields, and several assessments in Korea. Through these sections the authors provide information about theory of mathematics education and its practices. It is especially interesting due to the research being conducted in an oriental country, noted for its prominent achievements in PISA (Programme for International Student Assessment).

In the first part, the mathematics curriculum, its history and development focused specifically on the characteristics of mathematics textbooks used in current elementary schools, was analyzed. The second part was divided into five chapters on teaching and learning practices. And discussed both a general overview of the topic and a more specific view for several branches of mathematics: Geometry, Statistics, Reasoning and Problem Solving.

Finally, the third part discusses assessments conducted by teacher during mathematics instruction, by Schools, by the Office of Education, and by the state.

I believe that some aspects presented in this book could have important applications in teacher training and mathematics education. I consider that this last part could be interesting for teacher training. The authors present the criteria related to expertise in student evaluations that are required from a mathematics teacher. A reflection on factors that make up expertise on student assessments and the measures for improvement of teacher’s student assessments is given.
I am wondering if the good results achieved by this country in international assessments are a result of this methodological approach. When PISA results from 2012 are examined, Korean students’ mean scores are typically above the OECD average.

Gómez-Chacón, I. Mª
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid (UCM), España

Mathematical thinking: how to develop it in the classroom

Masami Isoda and Shigeo Katagiri
World Scientific Publishing Company
ISBN10: 9814350842
Price (tentative): 
£27 (paperback)
Short description: 

Mathematical thinking: how to develop it in the classroom, by Masami Isoda and Shigeo Katagiri, is the first volume in the series Monographs on Lesson Study for Teaching Mathematics and Sciences. This book has the potential to make a significant positive contribution to elementary education and the practice in the classroom.

MSC main category: 
97 Mathematics education
MSC category: 

Developing mathematical thinking is one of major aims of mathematics education. Mathematical thinking: how to develop it in the classroom, by Masami Isoda and Shigeo Katagiri, is the first volume in the series Monographs on Lesson Study for Teaching Mathematics and Sciences. This book has the potential to make a significant positive contribution to elementary education and the practice in the classroom.
Katagiri was the president of the Society of Mathematics Education for Elementary Schools in Japan. His life work has been devoted to analyze and classify mathematical thinking from 1960’s. Mathematical thinking consists on a proposal of mathematical ideas, methods, and attitudes which support the process of thinking mathematically. His proposal tries to enhance teachers’ questions for eliciting mathematical thinking from children.
The book consists of two parts. Part I, entitled Mathematical thinking: theory of teaching mathematics, explains Katagiri’s theory about how to develop children who learn mathematics for themselves according to the authors. Part II of the book, entitled Developing mathematical thinking with number tables: how to teach mathematical thinking presents 12 examples of lessons from the viewpoint of assessment.
The two parts are very different in their structure and content. The first part includes seven chapters. There is a preface of the book, and an introductory chapter by Isoda which is very helpful. Chapter 1 is referred to as Katagiri’s theory. In Chapter 2 the authors give arguments in favour of the development of mathematical thinking. In Chapter 3, it is explained the structure and a categorization of the components of mathematical thinking, which they name: Mathematical Methods, Mathematical Ideas, and Mathematical Attitudes. These components will be described in Chapters 4, 5 and 6. Finally in Chapter 7 the authors offer to the teachers a way for questioning to enhance the students. Questioning called “Hatsumon” in Japanese has three major usage in a mathematics in Japan. The first type of Hatsumon is relative to develop, recognize or reorganize mathematical knowledge, method and value. The second type is aimed to change phases of teaching in the whole classroom. And, finally the third type of Hatsumon is aimed to do the internalization of those two types of Hatsumon into children’s mind.
The second part of the book consists of 12 examples of lesson studies over the past 50 years (that develop and assess students’ mathematical thinking). These 12 chapters have a similar structure with sections: types of mathematical thinking to be cultivated, grade taught, preparation, overview of lesson process, worksheet, lesson process, summarization on the blackboard, evaluation and further development. I would like to note that from the 12 chapter-lessons, six of them are concerned with the arrangement of numbers on the tables, another are devoted to sums of numbers, two consider squares of numbers on the tables and the relationships among the numbers within the chosen squares, and the last two of them deal with the arrangement of multiples and common multiples.
As reviewer I would like to highlight two topics: Thinking mathematically and mathematical thinking in the curriculum and Mathematical Attitudes.
Thinking mathematically and mathematical thinking in the Japanese curriculum
The ideas relative to the main topic of this book are familiar to the majority of readers from the Western culture, as we can find in Polya, Schoenfeld, and Mason and colleagues writtings. For instance, the introductory Chapter 1 explains the pedagogical approach, which is referred to as ‘‘Problem Solving Approach’’. The approach is presented in five phases: Posing the Problem, Planning the Solution, Executing Solutions, Discussion (Validation and Comparison), and Summarization and Further Development. The reader can find resemblances to Polya’s (1945) description of the four steps of problem solving which, despite numerous extensions and critique (e.g., Schoenfeld 1985; Schoenfeld 1992) is still a major reference in instructional materials for teachers (e.g. NCTM 2000). However, I would like to underline the effort made by the authors to introduce the culture of Japanese teachers and Japanese teaching into English. Likewise, the reader would like to know more about the influence of cultural factors on the integration of these ideas. Finally, I consider that it is quite worth to adapt the solving problem ideas to a class of elementary school and also to offer the description of the 11 types of Mathematical Methods that are exemplified and discussed: inductive thinking, analogical thinking, deductive thinking, integrative thinking, developmental thinking, abstract thinking, simplifying, generalization, specialization, symbolization, and quantification and schematization. Although, it is missing in the book a closer match between the theoretical and the practical parts.
I am wondering if the good results achieved by this country in international assessments are a result of this methodological proposal. The PISA mathematics literacy test asks students to apply their mathematical knowledge to solve problems set in real-world contexts. To solve the problems students must activate a number of mathematical competencies as well as a broad range of mathematical content knowledge. When PISA results in 2012 are examined, Japanese students’ mean scores are typically above the OECD average on the top. The place of Japan in mathematics is the 7th with scores 536, In sciences Japanese position is 4th, 547 and reading position is 4th, 538 (34 OECD member countries). Also, in 2003 and 2009 Japan got a good the mathematics position (in 2003 position 4, 534 (30 OECD member countries) in 2009 position 9, 529 (34 OECD member countries)).
Part of the reason pupils do so well in Japan, according to the OECD's delegate director of education, Andreas Schleicher, is that they handle their confidence to achieve their potential. In Japan -- which ranked 7th overall -- more than 80% of students disagreed or strongly disagreed that they proposed problems were difficult, and 68% disagreed or strongly disagreed that they gave up easily when confronted to a problem.
In spite of Japanese students are interested in inquiry-based learning, whereas science teaching at the upper secondary level does not catch their interest. This fact could contribute to an understanding of why Japanese students in PISA show relatively low levels of positive attitudes toward science (Yasushi, 2009, 175). For example, in this country higher mean performance has lower average levels of mathematics interest. Interest in math remains low. In the latest survey, 38 percent of Japanese students said they were interested in the things they learn in math class. The figure is five points higher than the survey carried out in 2003, but it is 15 points lower than the average for students in the OECD. The increment of Japanese students who are interested in mathematics was a good trend, even though the percentage of attitudes, those feel pleasure, interest and motivation in solving mathematical problems remains low and it raises the need of the development of mathematical attitudes, as was outlined in this book.
Mathematical Attitudes
The Mathematical Attitudes considered in Chapter 6 are named: Objectifying, Reasonableness, Clarity and Sophistication. Here, it is important to note that the authors consider attitude as a ‘‘mindset’’, a mathematical disposition, where one examines the obtained answers and seeks for ‘‘better’’ ways to describe a situation.
While the attitudes toward mathematics have long been studied (Kulm, 1980, McLeod 1992, Di Martino & Zan, 2011; Hannula, 2002; Ruffell et al., 1998), the study of mathematical attitudes has been less thoroughly developed.
As early as 1969, Aiken and Aiken suggested two classical categories, attitudes toward science (when the object of the attitude is science itself) and scientific attitudes (when the object is scientific processes and activities, i.e., scientific epistemology), which were later adapted by a number of authors (Hart, 1989; NCTM, 1989; Gómez-Chacón, 2000) to mathematics and denominated attitudes toward mathematics and mathematical attitudes.
Although developments in attitudes have focused more on attitudes toward mathematics, I would like to underline the importance of developing mathematical attitudes as this book tries to promote. I missed in the book a major discussion about both categories and why they focus on one of the categories. In the following, consequently, this discussion will deal with the distinction that should be drawn, in teaching and learning, between attitudes toward mathematics and mathematical attitudes such as made in previous studies (e.g., Gómez-Chacón, 2011, p. 149).
Attitudes toward mathematics refer to the valuation of and regard for this discipline, the interest in the subject and the desire to learn it. They stress the affective component - expressed as interest, satisfaction, curiosity, valuation and so on – more so than the cognitive component. Mathematical attitudes, by contrast, are primarily cognitive and refer to the deployment of general mathematical disposition and habits of mind. Disposition refers not simply to attitudes but to a tendency to think and to act in positive ways. Students’ mathematical attitudes are manifested in the way they approach tasks such as flexible thinking, mental openness, critical spirit, objectivity and so on, which are important in mathematics (see NCTM Standard 10 (1989), for instance). Due to the predominantly cognitive nature of mathematical attitudes, to be able to be regarded as attitudinal, they must also comprise some affective dimension: i.e., a distinction between what a subject can do (mathematical disposition and habits of mind) and what a subject prefers to do (positive attitude toward mathematics). In 1992 Schoenfeld coined the term enculturation to mean that becoming a good mathematical problem solver may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. Enculturation is entering and picking up the values of community or culture (Schoenfeld, 1992, p. 340). According to Schoenfeld, students need a socialization process, to be imbued with certain habits of mind and mathematical attitudes such as those mentioned above. Consequently, it is incumbent upon teachers to create environments favouring the inquisitive spirit, pursuit of ideas, research and questioning associated with the practice of mathematics.
Bearing in mind, then, that attitude is defined to be a psychological tendency expressed as the evaluation of an object from the two categories of attitude specified. Chapter 6 of Isoda and Kataragiri book (p. 111-120) is dedicated to mathematical attitudes and in the p. 50 is listed items to assess the student behaviour. The reader would like to know more about the cognitive-emotional processes involved in evaluation of students when doing mathematics. In addition, apart of the mentioned attitudes, it is missing other essential mathematical attitudes in the setting of classroom as flexible thinking, critical spirit, visual thinking, inductive attitude, curiosity, perseverance, creativity, independence, systematization, cooperation and teamwork and how those are monitoring.
Since the authors say that this approach is extended in Japan and Korea, the reader would like to know more about the effect of this implementation, independent of the results from PISA.
Di Martino P. & Zan R. (2001). Attitude toward mathematics: some theoretical issues. Proceedings of PME 25 (Utrecht, Netherlands), vol.3, 351-358.
Di Martino, P., & Zan, R. (2011). Attitude towards mathematics: a bridge between beliefs and emotions. ZDM, 43(4), 471–482.
Frenzel, A.C.; Goetz, T.; Lüdtke, O.; Pekrun, R. and Sutton, R. E. (2009). Emotional Transmission in the Classroom: Exploring the Relationship between Teacher and Student Enjoyment. Journal of Educational Psychology, Vol. 101, pp. 705-716.
Gómez-Chacón, I. Mª (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics, 43 (2), 149-168.
Gómez-Chacón, I. Mª. (2000). Matematica emocional. Los afectos en el aprendizaje matematico (Emotional mathematics. Affectivity in mathematics learning). Madrid: Narcea.
Gómez-Chacón, I. Mª (2011). Mathematics attitudes in computerized environments. A proposal using GeoGebra. In L. Bu and R. Schoen (eds.), Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra, (147-170). Sense Publishers.
Kulm, G. (1980). Research on Mathematics Attitude. In R.J. Shumway (Ed.), Research in mathematics education (pp.356-387). Reston, VA, NCTM.
Hart , L. (1989). Describing the Affective Domain: Saying What We Mean, en McLeod, D.B. y Adams, V.M. (eds.). Affect and Mathematical Problem Solving (pp. 37-45). Springer Verlag.
Hannula, M. (2002). Attitude toward mathematics: emotions, expectations and values. Educational Studies in Mathematics, 49, 25-46.
McLeod, D.B. (1992). Research on affect in mathematics education: A reconceptualization. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning (pp.575-596). New York: Macmillan.
N.C.T.M. (1989). Curriculum and Evaluation Standards. Reston: NCTM.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
OECD (2010). PISA 2009 Initial Report: Learning Trends (Volume V), PISA, OECD Publishing.
OECD (2013). PISA 2012 Assessment and Analytical Framework Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing.
Polya, G. (1945/1988). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.
Ruffell, M., Mason, J. & Allen, B. (1998). Studying attitude to mathematics. Educational Studies in Mathematics, 35, 1-18.
Schoenfeld, A. (1985). Mathematical problem solving. New York, NY: Academic Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.) Handbook of Research on Mathematics Teaching and Learning. (pp. 334-389). New York: McMillan.
Yasushi, O. (2009). Comparison of Attitudes toward Science between Grade 9 and 10 Japanese Students by Using the PISA Questions and Its Implications on Science Teaching in Japan, paper presented at the QEG Meeting, Offenbach, Germany, 19-21 October.

Gómez-Chacón, I. Mª
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid (UCM), España

Complex Analysis In the Spirit of Lipman Bers

Rubí E. Rodríguez, Irwin Kra and Jane P. Gilman
Short description: 

This book is a second edition of a book, by the same authors, also published by Springer in 2007.
It contains some amount of new information. In particular a new section on historical remarks by
Ranjan Roy is included.
It covers all the usual topics on "Complex Variables Theory" and I found it very useful as an Undergraduate
and Graduate text in Mathematics.

URL for publisher, author, or book: 
MSC main category: 
30 Functions of a complex variable
MSC category: 

The book under review is a second edition of a book by the same authors
and with the same title, also published by Springer in 2007.
It contains some amount of new information.
In particular it has an interesting section
with historical remarks by Ranjan Roy as an Appendix to
Chapter 1. More details on the new material will be given below.

As is stated in Chapter 1, Section 2, the main goal of the book
is to prove the equivalence of eight statements
all of them seminal in the theory of functions of one complex
variable, starting with the Cauchy-Riemann
equations and finishing with the Runge theorem on
approximation of complex analytic functions by
polynomials. Of course the key to obtain those
equivalences is the Cauchy integral theorem, which is
obtained in Section 4.6.

In Section 1.3 of Chapter 1 the plan of the proof
of these equivalences is explained. The proof
of (1)=>(2) is presented in Chapter 2 but the proof
of (8)=>(1) that closes the chain, has to wait until
Chapter 7, almost 200 pages later.

The authors use differential forms and Green's theorem
to obtain, locally, the existence of a primitive for a holomorphic
function on a domain D in the complex plane, which is crucial in
the theory. As is written in Section 4.8.
<>. This other approach is outlined
as Appendix II in Section 4.8 (not having appeared in the previous
edition of the book) and is developed in
most books on the topic. Let me quote just two:
For example, Conway and Rudin's books (see References 7
and 30 in the text).

Once the equivalence between the eight statements
already mentioned is obtained, a certain amount of important
classical results on holomorphic functions of one complex
variable is obtained. In particular, the Riemann
Mapping Theorem and the Dirichlet problem
on the existence of harmonic functions on a disc with
prescribed values on the boundary are study in depth.

In relation with the Dirichlet problem, a study of subharmonic
functions and the Green function is carried out (this is new in
this edition) and the Dirichlet problem is revisited using
subharmonic functions and Perron's Principle. Riemann
factorization theorem and infinite Blaschke products are also

Another topic, new in this edition, is the study of the
ring of holomorphic functions on a domain. It is worth
remarking Theorem 10.12 which characterizes the ring
isomorphism between two proper domains of the extended complex

In spite of the book covering almost all the classical results
of which are known as "Complex Variables", a proof of Picard's
theorem (mentioned in Section 6.2) will have been a good
closing for the book but, unfortunately, it is not included.

The book is carefully written and
each chapter has an interesting list of exercises.
I found it very useful as am Undergraduate and Graduate Text in Mathematics.

José M. Ansemil
Departamento de Análisis Matemático. Universidad Complutense de Madrid, Spain

Analysis and Algebra on Differentiable Manifolds : a workbook for students and teachers

Pedro Martínez Gadea, Jaime Muñoz Masqué, and Ihor V. Mykytyuk
London: Sringer
Short description: 

The book contains a collection of solved problems on several topics of Differentiable Manifolds and their geometry.

MSC main category: 
53 Differential geometry

The book "Analysis and Algebra of Di erentiable Manifolds" contains a
collection of solved problems on diverse topics enclosed in the theory of di er-
entiable manifolds and their geometry. Although the text does not cover the
theory to which the problems refer, it includes introductory explanations at the
beginning of each section. In addition, appropriate references containing the
theory needed to solve the problems are pointed out. At the end of each chap-
ter other related references and further readings are suggested. It also contains
66 gures.
The book is organized as follows. The rst three chapters comprehend all
the topics included in an standard course on di erentiable manifolds at an un-
dergraduate level, starting from the very de nition of a di erentiable manifold,
passing through its tensor algebra, and ending with integration on manifolds
and de Rham cohomology. This rst part is of great value for both students
and teachers. The second part of the book is devoted to Lie groups, Rieman-
nian geometry and bre bundles, subjects that must be part of a di erential
geometer's background. The problems selected in this second part cover from
an advance undergraduate level to a Master level. The importance of these
chapters is that they treat topics for which signi cant examples and problems
are very difficult to nd in the literature. This collection of exercises allows the
reader to explore and to handle many concepts which otherwise could remain
just as theoretical objects.

Ignacio Luján
Universidad Complutense de Madrid