Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.     

Analysis of Pathogenic Bacterial and Yeast Biofilms Using the Combination of Synchrotron ATR-FTIR Microspectroscopy and Chemometric Approaches

Biofilms are assemblages of microbial cells, extracellular polymeric substances (EPS), and other components extracted from the environment in which they develop. Within biofilms, the spatial distribution of these components can vary. Here we present a fundamental characterization study to show differences between biofilms formed by Gram-positive methicillin-resistant Staphylococcus aureus (MRSA), Gram-negative Pseudomonas aeruginosa, and the yeast-type Candida albicans using synchrotron macro attenuated total reflectance-Fourier transform infrared (ATR-FTIR) microspectroscopy. We were able to characterise the pathogenic biofilms’ heterogeneous distribution, which is challenging to do using traditional techniques. Multivariate analyses revealed that the polysaccharides area (1200–950 cm⁻¹) accounted for the most significant variance between biofilm samples, and other spectral regions corresponding to amides, lipids, and polysaccharides all contributed to sample variation. In general, this study will advance our understanding of microbial biofilms and serve as a model for future research on how to use synchrotron source ATR-FTIR microspectroscopy to analyse their variations and spatial arrangements.     

Stable silencing of SNAP-25 in PC12 cells by RNA interference

Background  

SNAP-25 is a synaptic protein known to be involved in exocytosis of synaptic vesicles in neurons and of large dense-core vesicles in neuroendocrine cells. Its role in exocytosis has been studied in SNAP-25 knockout mice, in lysed synaptosomes lacking functional SNAP-25 and in cells after treatment with botulinum toxins A or E that specifically cleave SNAP-25. These studies have shown that SNAP-25 appears to be required for most but not all evoked secretion. In order to further study the role of SNAP-25 in catecholamine secretion from PC12 cells we have used the recently developed technique of RNA interference to generate PC12 cell lines with virtually undetectable levels of SNAP-25. RNA interference is the sequence-specific silencing or knockdown of gene expression triggered by the introduction of double-stranded RNA into a cell. RNA interference can be elicited in mammalian cells in a number of ways, one of which is by the expression of small hairpin RNAs from a transfected plasmid. Selection of stably transfected cell lines expressing a small hairpin RNA allows one-time characterization of the degree and specificity of gene silencing and affords a continuing source of well-characterized knockdown cells for experimentation.     

Long-term trend in CHF2Cl (HCFC-22) from high spectral resolution infrared solar absorption measurements and comparison with in situ measurements

The average tropospheric volume mixing ratio of CHF₂Cl (HCFC-22) has been retrieved from a time series of high spectral resolution ground-based infrared solar absorption spectra recorded with the McMath Fourier transform spectrometer located at the U.S. National Solar Observatory facility on Kitt Peak in southern Arizona (31.9°N, 111.6°W, 2.09 km altitude) for the time period October 1987–November 2002. The retrievals are based on fits to the well-isolated, unresolved 2ν₆ Q branch at 829.05 cm⁻¹ and the SFIT2 retrieval algorithm. The measured daily averages show a near linear rise per year in the mean tropospheric volume mixing ratio as a function of time with a best fit yielding an average increase rate of (5.66±0.15) parts per trillion (10⁻¹²) by volume per year, corresponding to (6.47±0.17)%yr⁻¹, 1 sigma, at the beginning of the time series. The tropospheric mixing ratios retrieved from the solar spectra have been compared with monthly average surface flask sampling measurements from the Climate Monitoring and Diagnostic Laboratory (CMDL) station at Niwot Ridge, Colorado (40.0°N, 105.5°W, 3013 m altitude), archived measurement from the same location, and early CMDL northern hemisphere Pacific cruise measurements. The average ratio of the retrieved tropospheric mixing ratio relative to the CMDL surface mixing ratio is 1.053 for the overlapping 1987 to 2002 time period. The retrieved mean tropospheric mixing ratio is consistent with the surface measurements within the errors estimated for the remote sensing observations.