Mathematics education in the 20th century: Ólafur Daníelsson’s impact

Abstract

The first Icelandic mathematician to complete a doctoral degree was Ólafur Daníelsson (1877–1957). He first completed a Mag. Scient. degree at the University of Copenhagen to be eligible to teach in the Danish high school system. His university professors were two wellknown geometers: Hieronymus Georg Zeuthen and Julius Petersen, a textbook writer. However, he was not hired as the mathematics teacher at the sole Icelandic high school when he returned home in 1904; the position went to an engineer. Daníelsson then began to prepare his doctoral thesis, which he defended in 1911: Nogle Bemærkninger om algebraiske Flader
der kunne bringes til at svare entydigt til en Plan Punkt for Punkt /Several remarks on algebraic surfaces which could have onetoone correspondence to a plane. The thesis was an extension of earlier works by professor Zeuthen and other wellknown scientists of the time: Clebsch, Castelnuevo and Cremona. Meanwhile, to support his growing family, Daníelsson taught pupils privately and composed a little arithmetic textbook, Reikningsbók/Arithmetic, published in 1906. Daníelsson was appointed as the first mathematics teacher at the Iceland Teacher College at its establishment in 1908. The students were mature people who may have been teaching for a number of years but who had not enjoyed schooling themselves. Daníelsson revised his textbook Arithmetic for their needs, republishing it in 1914. In 1919, a mathematics stream was established at Reykjavík High School following Daníelsson’s and his friends’ initiative. Daníelsson was appointed to develop it, aiming for students to become eligible to attend the Polytechnic College in Copenhagen and to take university studies in the sciences. Before that, students had to spend an extra year abroad. At that time Daníelsson began his mission to create high school mathematics textbooks. He wrote four textbooks in the 1920s, out of which two, the third version of Arithmetic and Algebra, became very influential through the midtwentieth century and shaped the mathematics education of generations. Eventually they were replaced in the 1970s by “New Math” textbooks. His two geometry textbooks, however, proved to be too ambitious for his students who had little previous experience with geometric concepts. Daníelsson’s approach to school mathematics was strictly academic. His teaching inspired his students, at least if they showed talent and commitment. One of them said, “What especially influenced us was his enthusiasm and respect for mathematics.” He explained mathematics in an intelligible way, supported by proofs if he thought it would be useful, but allowed space for initiative and creativity only in his verbal exercises. Daníelsson’s extensive influence may be attributed to several factors. His strong personality and firm belief in mathematics as a superb science made him an excellent champion of mathematics education. Moreover, he was the mathematics teacher of the first 167 primary school teachers in Iceland. His former students propagated Daníelsson’s vision and interpretation of school mathematics, and one of them, E. Bjarnason, published a primary school arithmetic textbook in 1927–1929 which has been deemed to be a simplified version of Daníelsson’s own Arithmetic for adolescents. Bjarnason’s textbook was chosen as the sole arithmetic textbook during the 1940s to 1970s when a state monopoly publishing house ensured the provision of free textbooks for all children. Bjarnason’s textbook may have been considered as the most suitable preparation for high school freshmen due to its compatibility with Daníelsson’s Arithmetic. Another factor was that Reykjavík High School was the only school of its kind. When another school became established in 1930, Daníelsson became protector of its mathematics and science stream. Admission to Reykjavík High School became restricted in 1929, which created strong competition. Following this, new lower secondary schools were established around the country in the 1930s for the common people who had not had the opportunity to attend school after age 14. Daníelsson’s Arithmetic was taken up in more and more of these schools to
enable their most promising students to transfer to one of the upper grades of the sixyear Reykjavík High School. In 1946, the sixyear high schools were reduced to fouryear upper secondary schools, and a national entrance examination was implemented in a number of lower secondary schools as a precondition for admission to these upper secondary schools. In deference to the hitherto dominant Reykjavík High School, the mathematics syllabus for the entrance examination was taken from the former second grade of this school, which included Daníelsson’s Arithmetic and Algebra textbooks. The syllabus and its exam remained in place from 1946 to 1976, although alternatives to Daníelsson’s textbooks were gradually brought in, especially after 1968 when the “New Math” was introduced in Iceland. Daníelsson retired from teaching in 1941 and died in 1957. His influence spanned nearly seven decades, from 1908 when he began teaching at Iceland’s Teacher College, until the 1976 when his textbooks were removed from the reading list of the national entrance examination. His legacy as a dedicated mathematician is unquestioned. Mathematics education in Iceland was shaped by his vision. The conditions in Iceland at this time, such as the restricted access to one of its two high schools, national isolation during the two world wars, the great depression between them, and finally national independence, created circumstances where a single policy dominated and there came to be no discussion (Kristín Bjarnadóttir, 2009). Many generations of teachers did not know other mathematics textbooks than those built on Daníelsson’s books. Because they were so genuine they were not debated and no discussion took place until long after his death.