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.

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 Nautical Almanac. 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.

The list of names from the 20th century is started with E.T. Bell, a popularizer of mathematics whose Men of mathematics 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 Mathematical Association of America (MAA) was established in 1915 as an offspring of the American Mathematical Society (AMS). The National Council of Teachers of Mathematics (NCTM) with the first president Charles Austin was founded in 1920 as a follow up for the Men's Mathematics Club 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 The education of T.C. MITS: what modern mathematics means to you. 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 math wars 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 New Math, 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 A beautiful mind by Sylvia Nasar and the eponymous film.

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 Space Race possible during the Cold War 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 New Math.

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.