Linear algebra proofs: ways of understanding and ways of thinking in the formal world

Research on teaching and learning of proofs in linear algebra is scarce. To help fill this gap, we interviewed six students taking a second linear algebra course and examined some of their coursework that they handed in. In this paper, we examined students’ ways of understanding and ways of thinking (Harel, 2008) in the formal world (Tall, 2008) of mathematical thinking from their statements as they unpacked a particular linear algebra proof. The results show that students were able to unpack and adjust the proof formally in a second course and reacted positively

A mathematician’s deliberation in reaching the formal world and students’ world views of the eigentheory

International audienceIn this paper we analyzed a mathematician’s journals of 5-day teaching episodes on eigenvalues and eigenvectors in a first-year linear algebra course, as well as his students’ responses to a survey. We employed Tall’s (2013) three world model, to follow the mathematician’s and his students’ movements between the three worlds. The study revealed that despite the mathematician’s efforts in demonstrating a more holistic view of the concepts, many students found linear algebra very abstract and gravitated more toward symbolic thinking

Developing a Machine Learning-Based Clinical Decision Support Tool for Uterine Tumor Imaging

Uterine leiomyosarcoma (LMS) is a rare but aggressive malignancy. On imaging, it is difficult to differentiate LMS from, for example, degenerated leiomyoma (LM), a prevalent but benign condition. We curated a data set of 115 axial T2-weighted MRI images from 110 patients (mean [range] age=45 [17-81] years) with UTs that included five different tumor types. These data were randomly split stratifying on tumor volume into training (n=85) and test sets (n=30). An independent second reader (reader 2) provided manual segmentations for all test set images. To automate segmentation, we applied nnU-Net and explored the effect of training set size on performance by randomly generating subsets with 25, 45, 65 and 85 training set images. We evaluated the ability of radiomic features to distinguish between types of UT individually and when combined through feature selection and machine learning. Using the entire training set the mean [95% CI] fibroid DSC was measured as 0.87 [0.59-1.00] and the agreement between the two readers was 0.89 [0.77-1.0] on the test set. When classifying degenerated LM from LMS we achieve a test set F1-score of 0.80. Classifying UTs based on radiomic features we identify classifiers achieving F1-scores of 0.53 [0.45, 0.61] and 0.80 [0.80, 0.80] on the test set for the benign versus malignant, and degenerated LM versus LMS tasks. We show that it is possible to develop an automated method for 3D segmentation of the uterus and UT that is close to human-level performance with fewer than 150 annotated images. For distinguishing UT types, while we train models that merit further investigation with additional data, reliable automatic differentiation of UTs remains a challenge

The Human Ecology and Geography of Burning in an Unstable Savanna Environment

According to new ecological theories, many savannas are inherently in disequilibrium and can flip from tree-dominated to grass-dominated landscapes depending upon the disturbance regime. In particular, a shift in a fire regime to a more frequent and intensive one can radically alter the tree-to-grass ratio in a given savanna. Drawing upon the ecological buffering model we argue that savanna persistence requires a relatively stable fire regime. We hypothesize that anthropogenic burning practices perform this function by producing a regular annual spatiotemporal pattern of fire that is linked to vegetation type. We test this hypothesis using a study of two areas, one in Mali and the other Burkina Faso. We use two sources of satellite data to produce an 11-year time series of the spatiotemporal pattern of fires and an example of the annual burned area pattern these fires produce. We combine the analysis of satellite imagery with interviews of rural inhabitants who set fires to understand the logic underlying the patterns of fire. Analysis of a time series of imagery reveals a strikingly regular annual spatiotemporal pattern of burning for both study areas, which cannot be explained by the regional climatic pattern alone. We conclude that the regularity of the annual fire regime in West Africa is a human-ecological phenomenon closely linked to vegetation type and controlled by people\u27s burning practices. We argue that the anthropogenic burning regime serves to buffer the savanna and maintain its ecological stability

Micellar structure and transformations in sodium alkylbenzenesulfonate (NaLAS) aqueous solutions: effects of concentration, temperature, and salt

We investigate the shape, dimensions, and transformation pathways of micelles of linear sodium alkylbenzenesulfonate (NaLAS), a common anionic surfactant, in aqueous solution. Employing Small Angle Neutron Scattering (SANS) and surface tensiometry, we quantify the effects of surfactant concentration (0.6–15 wt%), temperature (5–40 °C) and added salt (≤0.35 M Na2SO4). Spherical micelles form at low NaLAS (≤2.6 wt%) concentration in water, and become elongated with increasing concentration and decreasing temperature. Addition of salt reduces the critical micelle concentration (CMC) and thus promotes the formation of micelles. At fixed NaLAS concentration, salt addition causes spherical micelles to grow into cylindrical micelles, and then multilamellar vesicles (MLVs), which we examine by SANS and cryo-TEM. Above a threshold salt concentration, the MLVs reach diameters of 100 s of nm to few μm, eventually causing precipitation. While the salt concentrations associated with the micelle-to-cylinder transformation increase only slightly with temperature, those required for the cylinder-to-MLV transformation exhibit a pronounced, linear temperature dependence, which we examine in detail. Our study establishes a solution structure map for this model anionic surfactant in water, quantifying the combined roles of concentration, temperature and salt, at practically relevant conditions

Ancient pigs reveal a near-complete genomic turnover following their introduction to Europe

Archaeological evidence indicates that pig domestication had begun by ~10,500 y before the present (BP) in the Near East, and mitochondrial DNA (mtDNA) suggests that pigs arrived in Europe alongside farmers ~8,500 y BP. A few thousand years after the introduction of Near Eastern pigs into Europe, however, their characteristic mtDNA signature disappeared and was replaced by haplotypes associated with European wild boars. This turnover could be accounted for by substantial gene flow from local Euro-pean wild boars, although it is also possible that European wild boars were domesticated independently without any genetic con-tribution from the Near East. To test these hypotheses, we obtained mtDNA sequences from 2,099 modern and ancient pig samples and 63 nuclear ancient genomes from Near Eastern and European pigs. Our analyses revealed that European domestic pigs dating from 7,100 to 6,000 y BP possessed both Near Eastern and European nuclear ancestry, while later pigs possessed no more than 4% Near Eastern ancestry, indicating that gene flow from European wild boars resulted in a near-complete disappearance of Near East ancestry. In addition, we demonstrate that a variant at a locus encoding black coat color likely originated in the Near East and persisted in European pigs. Altogether, our results indicate that while pigs were not independently domesticated in Europe, the vast majority of human-mediated selection over the past 5,000 y focused on the genomic fraction derived from the European wild boars, and not on the fraction that was selected by early Neolithic farmers over the first 2,500 y of the domestication process

The 13th Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Ngā mihi aroha ki ngā tangata katoa and warm greetings to you all. Welcome to Herenga Delta 2021, the Thirteenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics. It has been ten years since the Volcanic Delta Conference in Rotorua, and we are excited to have the Delta community return to Aotearoa New Zealand, if not in person, then by virtual means. Although the limits imposed by the pandemic mean that most of this year’s 2021 participants are unable to set foot in Tāmaki Makaurau Auckland, this has certainly not stopped interest in this event. Participants have been invited to draw on the concept of herenga, in Te Reo Māori usually a mooring place where people from afar come to share their knowledge and experiences. Although many of the participants are still some distance away, the submissions that have been sent in will continue to stimulate discussion on mathematics and statistics undergraduate education in the Delta tradition. The conference invited papers, abstracts and posters, working within the initial themes of Values and Variables. The range of submissions is diverse, and will provide participants with many opportunities to engage, discuss, and network with colleagues across the Delta community. The publications for this thirteenth Delta Conference include publications in the International Journal of Mathematical Education in Science and Technology, iJMEST, (available at https://www.tandfonline.com/journals/tmes20/collections/Herenga-Delta-2021), the Conference Proceedings, and the Programme (which has created some interesting challenges around time-zones), by the Local Organizing Committee. Papers in the iJMEST issue and the Proceedings were peer reviewed by at least two reviewers per paper. Of the ten submissions to the Proceedings, three were accepted. We are pleased to now be at the business end of the conference and hope that this event will carry on the special atmosphere of the many Deltas which have preceded this one. We hope that you will enjoy this conference, the virtual and social experiences that accompany it, and take the opportunity to contribute to further enhancing mathematics and statistics undergraduate education. Ngā manaakitanga, Phil Kane (The University of Auckland | Waipapa Taumata Rau) on behalf of the Local Organising Committ

Navigating through the mathematical world: Uncovering a geometer’s thought processes through his handouts and teaching journals

International audienc

A mathematician’s deliberation in reaching the formal world and students’ world views of the eigentheory

International audienceIn this paper we analyzed a mathematician’s journals of 5-day teaching episodes on eigenvalues and eigenvectors in a first-year linear algebra course, as well as his students’ responses to a survey. We employed Tall’s (2013) three world model, to follow the mathematician’s and his students’ movements between the three worlds. The study revealed that despite the mathematician’s efforts in demonstrating a more holistic view of the concepts, many students found linear algebra very abstract and gravitated more toward symbolic thinking