A Gardner's workout. Training the mind and entertaining the spirit (Q2740532)

This is my favorite of the many Martin Gardner's books I have read and reviewed for Zentralblatt MATH. It has more open problems than any of Martin's books I know -- and most of these open problems are his own. The problems are classic: they are clearly posed for everyone to easily understand, and very challenging, destined to take a good chunk of life from successful conquerors! Here are a couple of representative examples:NEWLINENEWLINENEWLINEWhat is the minimal area of surfaces inside a transparent cube that will render it opaque? NEWLINENEWLINENEWLINECan the surface of a cube be covered, without overlap, with \(n\) congruent polygons, where \(n\) is any integer greater than 1?NEWLINENEWLINENEWLINEThe anthology consists of 41 chapters, all published in various academic journals and popular magazines after Martin Gardner's 25 glorious years of ``\textit{Mathematical Games}'' column at the \textit{Scientific American}. This is the first time these pieces are assembled in the form of a book. The subject matter covers a wide range of topics, including computer and calculator ``magic'' tricks; discussion of Gary Kasparov's defeat by Deep Blue in chess; chess puzzles; mathematical word play games; magic squares, tiling, covering, dissection, directed graphs and other combinatorial recreations; etc. Part two of the anthology consists of seven mathematical and math education book reviews.NEWLINENEWLINENEWLINEMartin Gardner dedicates his book ``To all the underpaid teachers of mathematics, everywhere, who love their subject and are able to communicate that love to their students.'' He cares about teachers and education so much, that for the first time in my memory he speaks up on secondary mathematical education in the United States, and takes on such an omnipotent organization as the National Council of Teachers of Mathematics (NCTM). The chapter is called ``Fuzzy New New Math'', and Gardner is reviewing three books, including the 1997 NCTM Yearbook. He writes:NEWLINENEWLINENEWLINE``Fuzzy-math teachers are urged by contributors to the yearbook to cut down on lecturing to passive listeners. No longer are they to play the role of `sage on stage'. They are the `guide on the side'. Classes are divided into small groups of students who cooperate in finding solutions to `open-ended' problems by trial and error. This is called `interactive learning'. \dots Getting a correct answer is considered less important than shrewd guesses based on insight, hence the term `fuzzy math'. Formal proofs are downgraded.''NEWLINENEWLINENEWLINE``Teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps, even proving, that the area of the largest square exactly equals the combined areas of the two smaller squares. According to fuzzy math, this is a terrible way to teach the theorem. Students must be allowed to discover it for themselves.''NEWLINENEWLINENEWLINEIndeed, I was encouraged when NCTM came up with its ``10 Commandments'' -- or 10 Standards -- in 1988. The first standard was seemingly taken straight from my 1987 book ``\textit{Mathematics as Problem Solving}''. I did not mind NCTM using my title without my permission -- indeed, I was happy that the most powerful organization armed itself with problem solving -- and I tried in my many talks for NCTM teachers to clarify what ``problem solving'' meant.NEWLINENEWLINENEWLINEBut Martin Gardner's observation that today ``formal proofs are downgraded'' can be immediately confirmed when one glances at NCTM web pages: \url{http://standards.nctm.org/document/eexamples/chap6/6.5/index.htm}NEWLINEOne does not see this politically incorrect word ``proof'' much; instead one encounters fuzzy-feely gentle terms as ``dynamic demonstrations'', ``visual `proofs' that require little or no symbolism or explanation''. The page I selected from the NCTM web site addresses presentation in classroom of the very same Pythagorean theorem that Martin mentions in his review. Let us read together:NEWLINENEWLINENEWLINE``\textit{Understanding the Pythagorean Relationship Using Interactive Figures}.NEWLINENEWLINENEWLINE\textit{The Pythagorean relationship}, \(a^2+b^2=c^2\) (\textit{where }\(a\) \textit{and }\(b\) \textit{are the lengths of the legs of a right triangle and }\(c\) \textit{is the hypotenuse}), \textit{can be demonstrated in many ways, including with visual ``proofs'' that require little or no symbolism or explanation. The activity in this example presents one dynamic version of a demonstration of this relationship. Visual and dynamic demonstrations can help students analyze and explain mathematical relationships}.''NEWLINENEWLINENEWLINEThe NCTM web discussion ends with an admission that shocks me:NEWLINENEWLINENEWLINE``\textit{Teachers should encourage their students to consider why it is important to repeat the demonstrations for different right triangles}.''NEWLINENEWLINENEWLINENCTM says that nothing at all has been proven by the demonstration, and we have to repeat the demonstration for \textit{each} right triangle! But there are so many of them!NEWLINENEWLINENEWLINETo paraphrase the Chrysler Chairman, NCTM ought to ``lead, follow or get out of the way.'' I do not mean here to follow our President's ``\textit{Teaching to the Test}'' slogan. I suggest that it is time to build our mathematics education around the essence of what mathematics is and what mathematicians do.NEWLINENEWLINENEWLINEI warmly recommend my favorite Martin Gardner book to everyone who enjoys mathematics, from secondary students to professional mathematicians and math educators. I would like to finish this review with my favorite quotation from the book:NEWLINENEWLINENEWLINE``The human mind is made of molecules, which are in turn made of atoms, which are turn made of electrons, protons, and neutrons. The protons and neutrons are made of quarks and electrons made of? Nothing except equations. Let's face it. You and I, at the lowest known level of our material bodies, are made of mathematics, pure mathematics, mathematics uncontaminated by anything else.''.