In both the United States and Europe, concerns have been raised about whether preservice and in-service training succeeds in equipping teachers with the professional knowledge they need to deliver consistently high-quality instruction. This article investigates the significance of teachers’ content knowledge and pedagogical content knowledge for high-quality instruction and student progress in secondary-level mathematics. It reports findings from a 1-year study conducted in Germany with a representative sample of Grade 10 classes and their mathematics teachers. Teachers’ pedagogical content knowledge was theoretically and empirically distinguishable from their content knowledge. Multilevel structural equation models revealed a substantial positive effect of pedagogical content knowledge on students’ learning gains that was mediated by the provision of cognitive activation and individual learning support.

The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.

Growth impairment in short stature homeobox-containing gene (SHOX) deficiency and Turner syndrome share a similar etiology. Because of the established effect of GH treatment on height in patients with Turner syndrome, we hypothesized that GH therapy would also stimulate growth in patients with SHOX deficiency. Our objectives were to evaluate long-term efficacy of GH treatment in short patients with SHOX deficiency and to compare the effect on final (adult) height (FH) in patients with SHOX deficiency and Turner syndrome. A prospective, multinational, open-label, randomized 3-arm study consisting of a 2-year control period and a subsequent extension period to FH. The treatment groups were 1) SHOX-D-C/GH (untreated during the control period, GH-treated during the extension), 2) SHOX-D-GH/GH, and 3) Turner-GH/GH (GH-treated during both study periods). Short-statured prepubertal patients with genetically confirmed SHOX deficiency (n = 49) or Turner syndrome (n = 24) who participated in the extension. Depending on the study arm, patients received a daily sc injection of 0.05 mg/kg recombinant human GH from start of the study or start of the extension until attainment of FH or study closure. Height SD score gain from start of GH treatment to FH was similar between the combined SHOX-deficient groups (n = 28, 1.34 ± 0.18 [least-squares mean ± SE]) and the Turner group (n = 19, 1.32 ± 0.22). In this FH population, 57% of the patients with SHOX deficiency and 32% of the patients with Turner syndrome achieved a FH greater than -2 SD score. GH treatment in short children with SHOX deficiency showed similar long-term efficacy as seen in girls with Turner syndrome.