In an expertise study with 94 mathematics teachers varying in their teaching experience (i.e., pre-service, induction, in-service teachers), we examined effects of teachers' professional knowledge and motivational beliefs on their ability to effectively integrate educational technology into their lesson plans. We assessed teachers' professional knowledge (i.e., content knowledge, pedagogical content knowledge, technological knowledge), and their motivational beliefs (i.e., self-efficacy, utility-value). Additionally, teachers developed a worked-out lesson plan for introducing the Pythagorean theorem to secondary students. They were explicitly instructed to refer to the integration of technology in their lesson plans. Experienced teachers considered more cognitively activating tasks in their lesson plans and exploited the potential of technology more than inexperienced teachers. Mediation analyses revealed that this effect was explained by teachers' perceived utility-value of educational technology but not by their professional knowledge. These findings suggest that teachers' motivational beliefs play a decisive role for effectively integrating technology in mathematics instruction.

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of "general arrangement varieties" and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.2010 Mathematics Subject Classification. 14L30, 14M25, 14J45. 1 2 J. HAUSEN, C. HISCHE, AND M. WROBELand that the elements t = (t 1 , t 2 , t 3 ) of the (three-dimensional) torus T = T 3 act on the points [z] = [z 0 , . . . , z 6 ] of P 6 via t · [z] = [z 0 , t 1 z 1 , t −1 1 z 2 , t 2 z 3 , t −1 2 z 4 , t 3 z 5 , t −1 3 z 6 ]. In particular, T acts diagonally. In order to link the situation in an optimal manner to toric geometry, we do a further step. Consider the torus T 6 ⊆ P 6 consisting of all points with only nonzero homogeneous coordinates and the splitting T 6 → T 3 × T 3 , t → (t 1 t 2 , t 3 t 4 , t 5 t 6 , t 1 , t 2 , t 3 ).

In an expertise study with 94 mathematics teachers varying in their teaching experience (i.e., pre-service, induction, in-service teachers), we examined effects of teachers’ professional knowledge and motivational beliefs on their ability to effectively integrate educational technology into their lesson plans. We assessed teachers’ professional knowledge (i.e., content knowledge, pedagogical content knowledge, technological knowledge), and their motivational beliefs (i.e., self-efficacy, utility-value). Additionally, teachers developed a worked-out lesson plan for introducing the Pythagorean theorem to secondary students. They were explicitly instructed to refer to the integration of technology in their lesson plans. Experienced teachers considered more cognitively activating tasks in their lesson plans and exploited the potential of technology more than inexperienced teachers. Mediation analyses revealed that this effect was explained by teachers’ perceived utility-value of educational technology but not by their professional knowledge. These findings suggest that teachers’ motivational beliefs play a decisive role for effectively integrating technology in mathematics instruction.

We classify all $\mathbb{Q}$ -factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.

We classify all Q-factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.