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"title": "Book review: “Pre-Calculus, Calculus, and Beyond” by Hung-Hsi Wu",
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"This\nis the sixth and final book of a series covering the K-12 curriculum, as\nan instrument for the mathematical education of school teachers. It is\nthe third and final volume of the series dedicated to high-school\nteachers. Unlike the two previous such volumes, which included topics\nthat had already been treated in the series (to ensure that high-school\nteachers could have at their disposal a set of self-contained\ninstruments for their mathematical education, expressly written for\nthem, thus not neglecting the pre-requisites to what they have to\nteach), this final book is composed of entirely new topics."
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"The first chapter is dedicated to trigonometry and the definition of\ntrigonometric functions with domain ",
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". It\nstarts with the basic definitions, the general notion of extension of a\nfunction, then applied to extending trigonometric functions, with the\nuse of the unit circle, to the interval ",
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".\nThe laws of sines and cosines, as well as\nother basic trigonometric identities, such as the addition formulas, are\nproven in this general setting. It proceeds to the definitions of radian\nand the new trigonometric functions obtained by switching from degrees\nto radians, and to the definition of polar coordinates. Finally,\ntrigonometric functions are put to use in the geometric interpretation\nof complex numbers and the derivation of the De Moivre and Euler\nformulas, with the exponential notation; applications are given to the\nstudy\nof ",
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"-th roots of unity, to a formulation of basic isometries\nin terms of complex numbers, and to the study of graphs of quadratic\nfunctions, with the use of rotations to eliminate the mixed term in\ngeneral quadratic equations in two variables. The chapter concludes with\nthe introduction of inverse trigonometric functions and a final section\nwhere the author analyzes the importance of these functions in the study\nof general periodic functions, which play a fundamental role in the\nphysics of many phenomena, through Fourier series. More advanced\ntreatments of trigonometric and general exponential functions are given\na brief overview, which provides adequate complementary useful knowledge\nto the readers."
]
},
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"content": [
"The following chapter proceeds with a rigorous treatment of real\nnumbers. Thus it becomes finally possible to justify what had previously\nbeen called “FASM” (the “fundamental assumption of school\nmathematics”) and enabled students to use real numbers, without\nbetraying the basic principles of mathematical studies, from the moment\nit becomes mandatory for the development of their mathematical\ninstruction, but before it is possible to include in the curriculum a\nrigorous treatment of the real line, due to the inner complexity of the\nsubject. After an algebraic reformulation of the theory of rational\nnumbers, the introduction of an extra axiom finally leads to the\nfundamental distinction of the sets of real numbers and rational\nnumbers, that can then be identified with a dense subset of the real\nline. The concept of limit of a sequence of real numbers is defined and\nits basic properties are then presented, proved and applied to the\nrigorous treatment of some of the concepts and properties\nthat had been previously accepted with the use of\nFASM, namely the existence\nand basic properties of positive ",
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"-th roots of positive real numbers and the\nfundamental theorem of similarity, followed by a whole chapter dedicated\nto a full study of the decimal expansion of a number, including\nrepeating and non-repeating decimals, and using the concept and basic\nproperties of infinite series."
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"A new chapter follows where the delicate concepts of length and area are\ntreated as rigorously as possible at this stage, based on a list of\nfundamental principles for geometric measurements that are accepted as a\nguide to the foundation of those concepts, but the inherent difficulties\nof these topics are explained. In this framework, the author introduces\nthe concept of rectifiable curve and identifies the problems one faces\nwhen trying to obtain a rigorous argument that leads to the formula for\nthe circumference of a circle, postponing the final solution to the end\nof the volume, where a more advanced treatment is given of trigonometric\nfunctions, that is put to this use.\n"
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"Some basic formulas for the area of elementary figures are revisited in\nthis more general setting and obtained using the assumptions of this\nchapter; a famous proof of the Pythagorean theorem using the concept of\narea is finally given a proper formulation, whereas it is very often\npresented to students without the due care to observe that it depends on\nrather subtle and nontrivial concepts and properties of area and without\nsome apparent geometrical properties being adequately proven. As the\nauthor explains in one of his illuminating pedagogical comments, this is\nanother example of how misleading some rather common incoherences\nin the teaching of school mathematics can be."
]
},
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"id": "p7",
"content": [
"After length and area, it is time for the introduction of some comments\non three-dimensional geometry and the concept of volume. By the\nformulation of some elementary principles that, at this stage, have to\nbe accepted without further foundation, the author proceeds to the proof\nof some basic facts on perpendicularity and parallelism of lines and\nplanes in three-space and to the analysis of Cavalieri’s principle,\nwhich leads to the formula for the volume of a sphere."
]
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"content": [
"The two final chapters are dedicated to an introduction of derivatives\nand integrals of real-valued functions of one variable and their basic\nproperties, and applications to trigonometric functions and to new\nformulations of the logarithmic and exponential functions; they start\nwith the notions of limit of a function in a point and of continuity.\n"
]
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"content": [
"As in the previous volume, this one also contains a very helpful Appendix\nwith a list of assumptions, definitions, theorems and lemmas from the\ncompanion volumes. I strongly recommend reading first the review of the\nfirst\nvolume (António de Bivar Weinholtz, ",
{
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"target": "https://dx.doi.org/10.4171/mag-56",
"content": [
"Book review, “Understanding numbers in elementary school mathematics” by Hung-Hsi Wu"
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",\nEur. Math. Soc. Mag. ",
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"content": [
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" (2021), pp. 66–67). There,\none can find the reasons why I deem this set of books a milestone in the\nstruggle for a sound mathematical education of youths.\nI shall not repeat here all the historical and scientific arguments that sustain this claim, but\nI have to restate,\nregarding this final volume, that although it is written for\nhigh-school teachers, as an instrument for their mathematical education\n(both during pre-service years and for their professional development),\nand to provide a resource for authors of textbooks, the set of its\npotential readers should not be restricted to those for which it was\nprimarily intended; it should include anyone with the basic ability to\nappreciate the beauty of the use of human reasoning in our quest to\nunderstand the world and the capacity and will to make the necessary\nefforts, which are required here as for any worthwhile enterprise.\nOf course, as the content and presentation of the three last volumes of the\nseries is of a more advanced nature, a wider mathematical background is\nrequired. This volume being the last of the series, we are now able to\nfully appreciate the magnitude of the enterprise undertaken by Prof. Wu\nand how it is indisputable, as I wrote before, that with this set of\nbooks at hand there is no excuse left for school (including high-school)\nteachers, textbook authors and government officials to persist in the\nunfortunate practice of trying to serve to school students mathematics\nin a way that is in fact unlearnable…"
]
},
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"type": "Paragraph",
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"content": [
"Like the previous two volumes, this one is punctuated with pedagogical\ncomments that give extremely useful advice regarding what content\ndetails should be used in classrooms and which are essentially meant to\nteachers; mathematical comments are also added to the main text, in\norder to extend the views of the reader whenever it helps to clarify the\nsubject in question. To the readers interested in the full scope of the\npedagogical comments of this volume I also recommend the lecture of my\npreceding\nreview (António de Bivar Weinholtz, ",
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",\nEur. Math. Soc. Mag. ",
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"125"
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" (2022), pp. 50–52), where\na detailed description is made of what the author considers to be the\nmain characteristics of mathematics and how they have been neglected in\nschools for such a long period of time and replaced by what he calls\n“Textbook School Mathematics” (TSM); the same concern is present in\nall the topics treated in the present volume."
]
},
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"content": [
"As\nit is almost inevitable in any printed book, there are some minor\nmisprints that can be easily detected and corrected by the reader.\nI just point out some details in formal definitions that deserve some\nattention."
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"The definition of the ",
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"-th term of a sequence (p. 118) as\nthe value assigned by the function (that the sequence is, by definition)\nto ",
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" is commonly found in these same terms in many mathematical\ntexts, but it can lead to some awkward consequences; for instance, it is\nthen not strictly true that “every sequence has infinitely many\nterms”, as the appreciation of this statement, with the above given\ndefinition, depends on the number of ",
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" (the set of natural\nnumbers) is infinite. With this definition, a ",
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" sequence\nwould have only one term…. A formal definition that would allow us\nto state that the number of terms of a sequence is always infinite, one\nfor each whole number, could be to identify the ",
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"-th term of the\nsequence ",
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"In the definition of convergence of regions (p. 230), apart from the stated condition on the boundaries, one needs some\nextra condition, as, for instance, the coincidence of the approximating regions with the limit region ",
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".\nThis condition is very easily verified in all the cases where Theorem 4.3 (convergence theorem for area) is applied in this book,\nand also in the graphical examples that are used in the treatment of area; this treatment,\nof course, has to rely on some intuitive assumptions at this stage."
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"Also, the\ndefinition of the limit in a point ",
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" (p. 286)\nadopted in this book is what we can call the “exclusive” limit,\ninasmuch as, to “test” the limit of the function, one only considers\nsequences in ",
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",\nas opposed to what we can call the “inclusive” limit\ndefinition, where we can also consider such sequences that can assume\nthe value ",
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"; but in the case of this “exclusive” limit, to\nensure that the limit is unique, when it exists, one has to assume as\nwell that ",
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"). It is not enough to ensure that it\nis just the limit of a sequence in ",
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" of ",
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",\ni.e.,\nif it is a limit point but not an accumulation point of the\ndomain, with this “exclusive” definition of limit, the function would\nhave every number as limit in ",
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" (because to contradict this fact\none would have to find a sequence in ",
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"; but this contradicts the\ndefinition of an isolated point). So, either one only considers domains\nwith no isolated points, or one has to define this kind of limit only in\naccumulation points and not in general limit points of the domain, as\nthe uniqueness of the limit is an essential feature of this concept.\nStrictly speaking, when considering the algebra of limits of functions,\nlike in Lemma 6.2 (p. 290), one also has to be careful to consider only\naccumulation points of the domain of the functions obtained by\nperforming each algebraic operation in the pair of functions, as it is\nnot mandatory that if a point is an accumulation point of the domain of\neach function in the pair it will also have this property with respect\nto the intersection of domains."
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"Finally, the definition of continuity\n(p. 289) is not affected by these subtleties, as it is not dependent on\nthe definition of limit of functions (only in the intuitive motivation\nof this concept a link is established with\nlimits). In\nfact, with the\nalternative (“inclusive”)\ndefinition of the limit of a function, to be continuous in a point of the domain could\nsimply be defined as having a limit in that point; however,\nif one aimed to use the adopted\n“exclusive” limit definition of continuity, one would have to treat\nseparately the isolated points of the domain. Nevertheless, this leads\nto the conclusion that in the proof of Lemma 6.3, on the “algebra of\ncontinuity”, one cannot fully rely on Lemma 6.2; once again we could be\nspared all these subtleties either if one excluded domains with isolated\npoints, or if one considered the “inclusive” definition of limit (in\nthis case, however, with some care also with domains in the algebra of\nlimits)."
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"content": [
"All these details should not, of course, be brought to a high school\nclassroom, although they can be of some use to teachers."
]
},
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"content": [
"As in the previous volumes of this series, on each topic the author\nprovides the reader with numerous illuminating activities, and an\nexcellent choice of a wide range of exercises."
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"Hung-Hsi Wu, ",
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". American Mathematical Society, 2020, 417 pages, Paperback ISBN\n978-1-4704-5677-1, eBook ISBN 978-1-4704-6006-8."
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"\nAntónio de Bivar Weinholtz is a retired associate professor of\nmathematics of the University of Lisbon, Faculty of Science, where he\ntaught from 1975 to 2009. He was a member of the scientific coordination\ncommittee of the new curricula of mathematics for all the Portuguese\npre-university grades (published between 2012 and 2014 and recently\nabolished). ",
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