We present the structural and optical characterization of colloidal PbSe nanocrystals. The lowest-energy exciton transitions in these structures, with diameters between 3 and 8 nm, occur at wavelengths between 1.0 and 1.85 µm. Band-edge luminescence spectra with quantum yields as high as 80% are observed. Unexpectedly long (∼300 ns) luminescence lifetimes are observed. These properties are promising for applications in optoelectronics and microscopy.

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This study identifies forms of interactions with diagrams that are involved in conjecturing; more specifically, how students display their thinking publicly through using multimodal representations. We describe how students interact with diagrams in both gestural and verbal forms, and examine how such multimodal interactions with diagrams reveal their reasoning about diagrams. We hypothesize that when limited information is given in a diagram, students make use of gestural and verbal expressions to compensate for those limitations as they engage in making conjectures. As a byproduct, the study also proposes a set of graphical representations of gestures that have been identified as important for geometrical reasoning. These can be employed to codify the gestural interactions and to depict the practices of teaching and learning in geometry classrooms.

We investigate experienced high school geometry teachers' perspectives on "authentic mathematics" and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving. This paper explores geometry teachers' perspectives on how the two-column proof form can be both a resource and a constraint in engaging students in proving. The two-column form has been used in the USA to teach students how to do proofs since early in the twentieth century (Herbst 2002a). But the two-column form has been criticized for perpetrating a formalistic image of proving, and calls for reform in the 1980s (NCTM 1989) had recommended that teachers decrease attention to two-column proving. In this article we examine a counterproposal that emerges from listening to teachers: namely, that the twocolumn form can actually be productive when engaging students in proving, by acting not only a constraint, but also a resource enabling complex and non-linear reasoning.

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