On the Mapper Algorithm: A study of a new topological method for data analysis

Mapper is an algorithm for describing high-dimensional datasets in terms of simple geometric objects. We give a new definition of Mapper, with which we are able to prove that Mapper is a functor and that Mapper is a homotopy equivalence for certain "nice" input data. To establish these results we describe the statistical theory of functorial clustering and the topological machinery of homotopy colimits. At the end of the document we show, by means of numerical experiments, that the functoriality of Mapper is useful in applications

Teacher feedback on procedural skills, conceptual understanding, and mathematical practices: A video study in lower secondary mathematics classrooms

Feedback is a prevalent teaching practice in mathematics classrooms, but few studies have documented how mathematics teachers enact feedback in classrooms. We investigated how 47 teachers provided feedback in 172 mathematics lessons in Norwegian lower secondary schools. We analyzed the quality of feedback, the quantity of feedback, and whether the feedback addressed students’ procedural skills, conceptual understanding, or engagement in mathematical practices. Teachers spent large amounts of time providing concrete and specific feedback, most of it addressing procedural skills while conceptual feedback was less common. The study highlights details of feedback relevant for both pre- and inservice mathematics teacher training

The instructional situations in which mathematics teachers provide substantive feedback

Feedback provided by mathematics teachers usually addresses procedural skills and, to a much lesser extent, other competencies such as conceptual understanding or engagement in mathematical practices. As most previous literature has studied feedback provided on homework or video prompts, how teachers provide such feedback in the classroom is poorly understood. Here, sixteen lessons taught by five teachers were purposefully sampled from a larger video study (172 lessons) as lessons with high-quality feedback according to a standardized observation instrument. The analysis focused on the instructional situations in which teachers provided feedback. When teachers provided procedural feedback, the situations were orderly and effective. Feedback on conceptual understanding and mathematical practices was provided in situations when students were especially challenged, and entailed a series of complex decisions, thereby placing demands on the teachers to manage both the students’ understanding and behavior. We argue that researchers should focus on how teachers and students negotiate the instructional situation to allow for feedback on conceptual understanding or mathematical practices