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Research continues to illustrate the important role of covariational and quantitative reasoning in the context of function and graphing. The same body of literature has emphasized that students and teachers often construct meanings for function and graphing that do not foreground these reasoning processes. In order to gain deeper insights into such meanings, we conducted clinical interviews with ten pre-service secondary teachers. In the present work, we illustrate the construct of shape thinking in relation to their graphing activity during the clinical interviews. We draw particular attention to the implications of shape thinking, including constraints generated by meanings rooted in such thinking, when confronted with noncanonical situations. [For the complete proceedings, see ED584443.]

It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus., Comment: To appear in Experimental Mathematics