Preservice teachers’ creation of dynamic geometry sketches to understand trigonometric relationships

Dynamic geometry software can help teachers highlight mathematical relationships in ways not possible with static diagrams. However, these opportunities are mediated by teachers' abilities to construct sketches that focus users' attention on the desired variant or invariant relationships. This paper looks at two cohorts of preservice secondary mathematics teachers and their attempts to build dynamic geometry sketches that highlighted the trigonometric relationship between the angle and slope of a line on the coordinate plane. We identify common challenges in the construction of these sketches and present examples for readers to interact with that highlight these issues. Lastly, we discuss ways that mathematics teacher educators can help beginning teachers understand common pitfalls in the building of dynamic geometry sketches, which can cause sketches not to operate as intended

Enacted task design: tasks as written in the classroom

This paper presents and describes the construct of enacted task design, which considers the way tasks are “written” (designed) by teachers. Two enactments by different teachers based on the same written algebra task were analyzed and compared using the math story framework (Dietiker, 2015). Variations in these stories highlight four dimensions of the teacher’s design work

Morphological assessment of the Octopus vulgaris species complex evaluated in the light of molecular-based phylogenetic inferences

Cryptic species are common in the ocean, particularly among marine invertebrates such as octopuses. Delineating cryptic species is particularly problematic in octopus taxonomy where the plasticity recorded among taxonomic characters often results in low resolution at the species level. This study investigated the morphological relationships among seven phylogenetic clades (identified using cytochrome c oxidase subunit I) of the broadly distributed Octopus vulgaris species complex and close relatives. Morphological analyses in this study were successful in delimiting O. sinensis, Brazilian O. vulgaris and O. vulgaris sensu stricto, which was congruent with the molecular findings of this study. Analyses based on male morphology were successful in distinguishing 14 of 15 total pairwise comparisons and proved to be a more reliable indicator of species-level relationships in comparison with female morphology. The majority of characters with the greatest discriminatory power were male sexual traits. Significant morphological differences were also recorded among sampling localities of conspecifics, with phenotype showing correlation with local environmental data. The findings of this study support the hypothesis that multiple O. vulgaris-like species are currently being incorrectly treated under a single species name, O. vulgaris. Octopuses being exported globally under the name O. vulgaris are of extremely high fisheries market value and profile. Our findings have potentially significant implications for the naming and conservation of commercially harvested members of this species complex throughout their ranges.Preprint2,793

The use of concept maps to assess preservice teacher understanding: a formative approach in mathematics education

The research reported in this article explored the methods by which concept maps served as formative assessment by capturing changes in the ways preservice mathematics teachers represented their understanding of algebra. The participants were enrolled in a course on high school algebra for teachers and created the maps on the first and last day of the course. Selected concept maps are presented to illustrate the methodology and results of the concept map analyses. Suggestions for ways instructors can use these analy- ses to make formative decisions are offered.Accepted manuscrip