ORIGINAL ARTICLE Future mathematics teachers’ professional knowledge of elementary mathematics from an advanced standpoint Nils Buchholtz • Frederick K. S. Leung • Lin Ding • Gabriele Kaiser • Kyungmee Park • Bjo ¨rn Schwarz Accepted: 17 September 2012 / Published online: 4 October 2012 Ó FIZ Karlsruhe 2012 Abstract This paper reports a joint research project by researchers from three countries on an international com- parative study that examines the professional knowledge of prospective mathematics teachers in elementary mathe- matics from an advanced standpoint. For this study, mathematical problems on various topics of elementary mathematical content were developed. Using this instru- ment, the mathematical knowledge of future teachers from Germany, Hong Kong, China (Hangzhou) and South Korea was measured empirically. The paper presents the design of the study, and also results are discussed. The results show that there are systematic differences among the participating countries; for example, the Korean future teachers outperform their counterparts in other countries. A more detailed analysis of the results suggests that the future teachers often do not seem to be able to link school and university knowledge systematically and cannot achieve the crucial ‘‘advanced standpoint’’ from the teacher training programme. 1 Introduction After taking up their university study when they attend the traditional introductory courses of university mathematics, many first-year student teachers realize the big gap between their ideas of the professional requirements of the mathematics teacher, on the one hand, and the advanced academic mathematics, on the other hand. The academic courses lack provision of learning opportunities for future teachers to sufficiently acquire the knowledge and skills for constructing a didactical ‘‘bridge’’ between university mathematics and student-oriented elementary mathematics. As a consequence, at least in Germany, there are high dropout rates amongst first-year student teachers, which may be an indication that the connection among the dif- ferent areas of teacher education (namely mathematics, mathematics education, and general education) is critically regarded as too weak (for an overview, see Blo ¨meke 2004). The general criticism of the absence of practical content in the teaching of mathematics at universities is not new and can be found in the ground-breaking book Elementary Mathematics from an Advanced Standpoint by Felix Klein, who as early as the beginning of the twentieth century described the following phenomenon, commonly known as the ‘‘double-discontinuity’’: ‘‘The young university student found himself, at the outset, confronted with problems that did not suggest, in any particular, the things with which he had been concerned at school. Naturally he forgot these things quickly and thoroughly. When, after finishing his course of study, he became a teacher, he suddenly found himself expected to teach the traditional ele- mentary mathematics in the old pedantic way; and, since he was scarcely able, unaided, to discern any connection between this task and his university mathematics, he soon fell in with the time honored way of teaching, and his university studies remained only a more or less pleasant memory which had no influence upon his teaching.’’ (Klein 1932, p. 1) N. Buchholtz (&)  G. Kaiser  B. Schwarz Faculty of Education, Psychology and Human Movement, University of Hamburg, Hamburg, Germany e-mail: Nils.Buchholtz@uni-hamburg.de F. K. S. Leung  L. Ding The University of Hong Kong, Hong Kong SAR, China K. Park Hongik University, Seoul, Korea 123 ZDM Mathematics Education (2013) 45:107–120 DOI 10.1007/s11858-012-0462-6