European Mathematical Society - 01A80
https://euro-math-soc.eu/msc-full/01a80
enRepublic of Numbers
https://euro-math-soc.eu/review/republic-numbers
<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 twenty short biographical chapters it is sketched how the role of mathematics in the American society and its educational system has evolved from the early 19th till the late 20th century. That is one chapter per decade, but the life span of the individual mathematicians is of course wider: From Nathaniel Bowditch (1773-1838) to John Nash (1928-2015). In the 19th century, the US were expanding and fighting for independence. Importing slaves was gradually abolished which entailed a civil war between the Northern and Southern states. In the 20th century they participated in global conflicts and survived a cold war. Around 1800, there were only nine colonial colleges, (for white men only), and they mainly trained lawyers, physicians, and clergy. In rural regions teaching to read and write was for the lucky ones and it was forbidden to teach slaves. In the 1990's, there are numerous renowned universities and a regular educational system was established, with mathematics taking an important place at all levels of education. How did this come about? That is what Roberts is illustrating with this selection of 23 biographical sketches (some chapters treat two persons simultaneously). He did not take the leading mathematicians to illustrate the evolution (only few were famous) but there is a diversity of characters and people who were in some sense related to mathematics, and often they were involved in educational issues.</p>
<p>
Here are some names from the first of these two centuries. Simple calculations were sufficient for every day life in 1800, except for navigation which required some knowledge of celestial mechanics. Nathaniel Bowditch taught himself mathematics which he needed as a sailor and wrote a book on navigation and later translated work of Laplace. Sylvanus Taylor had some education when entering the military. Later he became the director of West Point, the US military academy that he modeled after the Ecole Polytechnique in Paris and whose alumni played an important role in professionalizing mathematics in other places. Abraham Lincoln did not become a mathematician, but in his youth, he maintained a scrap book with elementary mathematical problems. Only some of its pages have been recovered. Catherine Beecher and Joseph Ray were authors of popular math text books, and Daniel Hill was a popular educator at West Point. J.W. Gibbs became famous as a mathematical physicist with his work on thermodynamics. Charles Davis was a naval officer who supervised the computation of the <em>Nautical Almanac</em>. and was later superintendent of the Naval Observatory. After the civil war (1861-1865), the educational system became more tolerant for women, Christine Ladd was one of the first women to become a researcher at John Hopkins University. She fulfilled all the requirements for a PhD but it was only awarded 44 years later in 1927 when she turned 80. Kelly Miller is an example of an African American who attended the "black" Howard University, and wrote a math textbook and essays on popular mathematics. H. Hollorith, known from the punch cards named after him, was also founder of the Tabulating Machine Company, which later grew into IBM and E.H, Moore is a mathematician known for several things like the Moore-Penrose inverse. He had some students that became famous mathematicians: G. Birkhoff, L. Dickson, and O. Veblen. Those names bring us to the end of the 19th century, with data processing on the horizon and mathematics and mathematicians being imported from Europe on a larger scale raising mathematics to a higher level.</p>
<p>
The list of names from the 20th century is started with E.T. Bell, a popularizer of mathematics whose <em>Men of mathematics</em> became a classic. By this time, education had been formalized. Classes were split according to the age of the pupils, lessons were separated by a bell signal, and schools had a non-teaching management. The <em>Mathematical Association of America</em> (MAA) was established in 1915 as an offspring of the <em>American Mathematical Society</em> (AMS). The <em>National Council of Teachers of Mathematics</em> (NCTM) with the first president Charles Austin was founded in 1920 as a follow up for the <em>Men's Mathematics Club</em> of the greater Chicago area. Edwin B. Wilson was a student of Gibbs and became mainly involved with statistics. The couple Liliane and Hugh Lieber are known for their series of booklets popularizing math and science with text in free verse format for easy reading by Liliane (maiden name Rosanoff, an emigrate from Ukraine) and drawings by Hugh. Their best known title is <em>The education of T.C. MITS: what modern mathematics means to you</em>. T.C MITS stands for The Celebrated Man In The Street. With WW II, computers came into vision and Grace Hopper designed a computer language that was a precursor of what later became COBOL. Izaak Wirszup studied mathematics under Zygmund in Poland, and survived a Nazi concentration camp. Zygmund, who had escaped the Nazis, invited him to the US where Wirszup became mainly involved in math education. The 1960's was the period where African Americans were fighting racial segregation and Edgar L. Edwards, Jr., was one of the first black teachers at the University of Virginia. Also Joaquin Diaz, although an American citizen from Puerto Rico, was subject of racial discrimination because he was considered Hispanic, and non-American. As an applied mathematician working on fluid dynamics, he was involved in NACA (precursor of NASA). The <em>math wars</em> of the 1980's was the fight over traditional versus "new" mathematics that was abruptly introduced in the US, a reform supported by the NCTM. Frank Allen, who was a believer in the original ideas of <em>New Math</em>, and who had been involved in NCTM became an active polemicist in the debate. The last man in the row is John Nash whose life is well known because of the biography <em>A beautiful mind</em> by Sylvia Nasar and the eponymous film.</p>
<p>
This enumeration of names shows that Roberts is not focussing on mathematical research at university level, but rather at the historical evolution of mathematical education at a lower level, which is of course not independent of what happens at the universities. Why these names? I guess any list of names can be criticized, but I think Roberts chose a good mixture of sex and race, that somehow represents how political and social circumstances have influenced the mathematical education. In the beginning, navigation and the military interest were stimulations for doing mathematics. The military definitely remained to have an important influence and WW II has given a boost to the development of math and science in the US because of the many scientists that fled Europe for the Nazis, which made a <em>Space Race</em> possible during the <em>Cold War</em> period. The latter events are however more important at a research level, and that is not so present in this book. Nevertheless the USSR having Sputnik first is related to the forcing initiative to introduce the <em>New Math</em>.</p>
<p>
Roberts has assigned one particular year to every chapter. Each chapter starts with an epigraph and the description of a particular event that happened in that year to the person that is going to be discussed. That introduction takes only one to three pages and should serve as an appetizer for the longer biography that is following. There is some discussion of the mathematics but it is nowhere technical (no formulas), and there is a photo of each of the mathematicians discussed (except for Abraham Lincoln who is not a mathematician anyway). There are notes and references for what is mentioned in the text but no extensive list for further reading. The book is a very readable survey that will be of interest to any mathematician and non-mathematician alike, but maybe more so for those who are particularly interested in the history of math education.</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 20 short biographical chapters, this book illustrates how the US evolved from the early 19th century with schools where children learned to read and write while mathematics was mainly of interest to navigators and astronomers to the end of the 20th century where mathematics had become a main ingredient at all levels of education.</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/david-lindsay-roberts" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Lindsay Roberts</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/john-hopkins-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Hopkins 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">9781421433080 (hbk), 9781421433097 (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">USD 29.95 (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">252</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/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://jhupbooks.press.jhu.edu/title/republic-numbers" title="Link to web page">https://jhupbooks.press.jhu.edu/title/republic-numbers</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li></ul></span>Fri, 20 Dec 2019 14:36:53 +0000Adhemar Bultheel50113 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/republic-numbers#commentsMillions, Billions, Zillions
https://euro-math-soc.eu/review/millions-billions-zillions
<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>
Journalists reporting on somebody else's results may easily be mistaken in citing the numbers that are not theirs, Perhaps authors change numbers on purpose to bias their arguments. In such cases unprepared and naive readers are easily deceived. In this book you can learn how to defend yourself from mistakes as an author and from deceptions as a reader.</p>
<p>
Kernighan is the guide on a tour where he shows all the number traps that people can easily fall into. There are of course the big numbers like millions and billions mentioned in the title. It is hard to get a mental idea of what they actually mean. As a consequence an interchange of millions and billions is a mistake easily made without being noticed. One should also be aware of what the really big numbers actually stand for when they are indicated by prefixes like mega, giga, tera, and peta. Even exa, zetta, and maybe yotta can come into the picture. In modern texts these terms regularly occur referring to the large amounts of digital data stored in for example the Deep Web. At the other end of the spectrum we should know something about micro, nano, pico, femto, atto etc. when reading about the tiny parts of the hardware on which these data are stored, or even further down the scale when reading about the Theory of Everything where the natural playground is at a subatomic level. Like in mathematics the super large and the super small often match to keep everything within finite boundaries.</p>
<p>
To avoid errors of course the units have to be right. Mistaking barrels for gallons, years for months, or hours for seconds may give quite unexpected and hard to believe interpretations of the numbers that are on display. Besides mistakes in scales, there is the extra complication that there are different systems of units like American miles, gallons, or degrees Fahrenheit that should somehow match with European kilometres, litres and degrees Celsius. Even a mile can have many different meanings in different contexts. All this requires carefully introducing the proper conversions. Just picking up a number from a website may easily lead to such mistakes if the proper conversion is not made. Another typical mistake is to confuse a square mile and a mile squared. This means that you should be aware that if you double the length of the sides of a square you get a surface four times as large. For a volume it is even more dramatic because that will give a volume or mass that is eight times as large.</p>
<p>
The previous examples are possible sources of mistakes. How should we detect them and how could we protect ourselves against malicious attempts to deceive us? We could compare different sources or different ways of computing. When the results are approximately the same we can probably trust the numbers. It is of course useful to check with some numbers from your own experience or numbers that you know like for example the population of your country. Approximate rounded values for these numbers can be used in crude and simple computations to get at least an idea about the magnitude of the number that should be expected. If the number presented to you deviates considerably, either it is fake or your computations are wrong. For example <em>Little's Law</em> is applicable for a simple check in many situations. Given the population of your country and an average human life span, Little's Law allows you to estimate for example the number of people that will turn 64 twenty years from now.</p>
<p>
Usually numbers appear in non-scientific texts with approximate rounded values. If one spots a specious number that is given with many digits, then it is probably the result of a computation or conversion and only the most significant digits or rounded values are actually meaningful. They are probably the result of some conversion, like a mountain over 4.000 meter high should not be referred to as over 13.123 feet. Statistics is another possible source of deception: the median is not the average, a correlation does not imply causation. It might also be interesting to know who did the statistics and the sampling or polling. Results may be consistently biassed towards the results of which some lobbyist or pressure group wants to convince you. Another well known trick is to fiddle with the scales used in the graphical representation of the numbers in pie or bar charts. Numbers representing a percent are again possible pitfalls that can put you on the wrong foot. A percent is definitely different from a percent point and you should also be aware that a percent increase is computed in terms of the lower number, while a percent decrease is referring to a percent of the larger number: a 50% decrease can only be compensated by a 100% increase.</p>
<p>
Kernighan gives many examples of all these issues, mostly from newspapers and websites. He also keeps his readers alert by continuously pushing them to do some mental calculation to estimate some results for themselves. As some kind of a test at the end of the book he gives many such problems that one should be able to approximately solve (he also gives his own estimates): How many miles did Google drive to get the pictures for Street View (for your country)? How long did that take? How much did it cost? Or, if you have a garden with some trees in it, how many leaves do you have to rake every autumn? And there are many of these so called <em>Fermi problems</em> throughout the book. Kernighan gives some tricks to solve them, hence the "test" at the end. However it certainly requires a lot of practising and training which the reader has to do for him or herself to acquire some routine in this,</p>
<p>
In this time of "fake news" and in a society that is more and more spammed by numbers, it seems like problems of numeracy among a general public is gaining interest and public awareness. More books devoted to different aspects of this issue seem to appear lately. Among the earlier examples are the books by John Allen Paulos <em>Innumeracy</em> (1988) and <em>A mathematician reads the newspaper</em> (1996). Kernighan mentions them in his survey of "books for further reading". However several more books appeared since 2010. Almost simultaneously with this one I received <em>Is This a Big Number?</em> (2018) by Andrew Elliott which is also reviewed <a href="/review/big-number" target="_blank">here</a>. Elliott has a more positive approach to the problem: how should I interpret the numbers presented to me (assuming they are correct) while Kernighan is more defensive: how to to stay away from mistakes or deceiving numbers.</p>
<p>
It should also be noted that Kernighan uses American units most of the time, and his examples are mostly related to the American situation, or from the American newspapers. His advise is of course generally applicable, but it might be an extra hurdle to take for European readers who are used to the metric system of metre-litre-gram. It is surprising that Kernighan does not discuss the difference between the short scale (a billion is $10^9$) and long scale (a billion is $10^{12}$) hence also missing the milliard ($10^9$) and billiard ($10^{15}$) and giving different meaning to trillions, and nomenclature higher up. These are also obvious ways to get the wrong numbers cited.</p>
<p>
This book does not need mathematics to read and it is actually not about mathematics at all. There is not a formula made explicit, even though the rule of 72 is explained (it takes 72/x units of time to double your capital when it has a compounding interest of x percent) and an idea is given about what it means to grow exponentially or by powers of 10 or powers of 2. The style of Kernighan is fluent and casual, but not particularly funny. The charm sits in his continuous teasing to make you think of these Fermi problems, and of course in his ample illustrations of how often authors are mistaken in citing numbers and how easily a reader can be deceived.</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 about numeracy learns you how to defend yourself against making mistakes with numbers or to recognize incorrectly cited numbers by explaining what the possible sources of these errors are. On the positive side, one learns how to solve Fermi problems, that is to make rough numerical estimates of certain quantities, using little available data and where the few computations can me made "on the back of an envelope".</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/brian-w-kerninghan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian W. Kerninghan</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton 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">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">978-0-691-18277-3 (hbk), 978-0-691-19013-6 (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">£ 17.99 (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">176</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://press.princeton.edu/titles/14171.html" title="Link to web page">https://press.princeton.edu/titles/14171.html</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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</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/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li></ul></span>Wed, 24 Oct 2018 12:08:48 +0000Adhemar Bultheel48773 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/millions-billions-zillions#comments