The notion of motion: covariational reasoning and the limit concept

This paper investigates outcomes of building students’ intuitive understanding of a limit as a function's predicted value by examining introductory calculus students’ conceptions of limit both before and after instruction. Students’ responses suggest that while this approach is successful at reducing the common limit equals function value misconception of a limit, new misconceptions emerged in students’ responses. Analysis of students’ reasoning indicates a lack of covariational reasoning that coordinates changes in both x and y may be at the root of the emerging limit reached near x = c misconception. These results suggest that although dynamic interpretations of limit may be intuitive for many students, care must be taken to foster a dynamic conception that is both useful at the introductory calculus level and is in line with the formal notion of limit learned in advanced mathematics. In light of the findings, suggestions for adapting the pedagogical approach used in this study are provided.     

DIBAL-mediated reductive transformation of trans-dimethyl tartrate acetonide into ε-hydroxy α,β-unsaturated ester and its derivatives

Stepwise, selective DIBAL reduction of the acetonide diester derived from tartaric acid followed by the Horner-Emmons reaction effectively provided desymmetrized hydroxy mono-olefination products in a one-pot operation.     

The relative detectability for mice of gaps having different ramp durations at their onset and offset boundaries

The effect on gap detectability of varying noise fall time (FT) and rise time (RT) of the gap boundary ramps was examined in mice using reflex modification audiometry, measuring inhibition of acoustic startle reflexes by variously shaped gaps just preceding reflex expression. In experiment 1 (n = 12) inhibition increased up to near-asymptotic values with longer FT (0, 1, 2, 3, 5, or 10 ms) and QT (quiet time, 0 to 13 ms), with a 2:1 trade-off between FT and QT. In experiment 2 (n = 24) inhibition increased for any RT above 0 ms (2, 3, 5, or 7 ms) if QT= 1 ms, but diminished with increased RT when QT = 3 or 8 ms. Enhanced detectability for subthreshold gaps by longer ramps results from their extending the apparent gap duration. The negative effect of increased RT for threshold gaps suggests the importance for gap detection of the stronger neural responses to sharp edges at the end of the gap shown previously in the mouse inferior colliculus. These effects are specific to gaps: inhibition for fixed (70-dB SPL) or varied level pulses (30 to 60 dB) was unaffected by varying the ramped edges (experiments 3 and 4, n = 9).     

An alternative to ratio method in sample surveys

Summary This paper examines a simple transformation which enables the use of product method in place of ratio method. The convenience with the former, proposed by Murthy [3], is that expressions for bias and mean square error (mse) can be exactly evaluated. The optimum situation in the minimum mse sense and allowable departures from this optimum are indicated. The procedure requires a good guess of a certain parameter, which does not seem very restrictive for practice. Two methods for dealing with the bias of the estimator are mentioned. An extension to use multiauxiliary information is outlined.     

Using the Knowledge Quartet to quantify mathematical knowledge in teaching: the development of a protocol for Initial Teacher Education

This study examined trainee teachers' mathematical knowledge in teaching (MKiT) over their final year in a US Initial Teacher Education (ITE) programme. This paper reports on an exploratory methodological approach taken to use the Knowledge Quartet to quantify MKiT through the development of a new protocol to code trainees' teaching of mathematics lessons. This approach extends Rowland's et al. work on the Knowledge Quartet (KQ). Justification for using the KQ to quantify MKiT, and the potential benefits such an attempt might provide those involved with ITE, are discussed. It is suggested that quantified MKiT data based on the Knowledge Quartet can be used to consider MKiT development in novice teachers in order to inform ITE programmes and form new theoretical loops between theory and practice in teacher education.