Fractions as Subtraction: An Activity‐Oriented Perspective from Elementary Children

A sample of third‐, fourth‐, and fifth‐grade student responses to the question “What is a fraction?” were examined to gain an understanding of how children in upper elementary grades make sense of fractions. Rather than measure children's understanding of fractions relative to mathematically conventional part–whole constructions of fractions, we attempted to understand children's actions and processes. A small but nontrivial group of children used subtraction (takeaway and removal) as a framework for understanding how fractions were created and written. An analysis of the content of their responses as well as a comparison of the performance of these children with that of children who used other ways of describing fractions suggests that the use of subtraction may be a reasonable (or at least not harmful) way for children to begin to access concepts related to fractions. Also, this study suggests that attention to children's understanding through the lens of children's activity might reveal ways of thinking and insights that are masked when we compare children's thinking in more structured research settings.     

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.     

Fine tuning Nesterov’s steepest descent algorithm for differentiable convex programming

We modify the first order algorithm for convex programming described by Nesterov in his book (in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, 2004). In his algorithms, Nesterov makes explicit use of a Lipschitz constant L for the function gradient, which is either assumed known (Nesterov in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, 2004), or is estimated by an adaptive procedure (Nesterov 2007). We eliminate the use of L at the cost of an extra imprecise line search, and obtain an algorithm which keeps the optimal complexity properties and also inherit the global convergence properties of the steepest descent method for general continuously differentiable optimization. Besides this, we develop an adaptive procedure for estimating a strong convexity constant for the function. Numerical tests for a limited set of toy problems show an improvement in performance when compared with the original Nesterov’s algorithms.     

Modelling alcohol problems: total recovery

Binge drinking in the UK is an increasing problem, resulting in negative health, social and economic effects. Mathematical modelling allows for future predictions to be made and may provide valuable information regarding how to approach solving the problem of binge drinking in the UK. We develop a 3-equation model for alcohol problems, specifically binge drinking, which allows for total recovery. Individuals are split into those that are susceptible to developing an alcohol problem, those with an alcohol problem and those in treatment. We find that the model has two equilibrium points: one without alcohol problems and one where alcohol problems are endemic in the population. We compare our results with those of an existing model that does not allow for total recovery. We show that without total recovery, the threshold for alcohol problems to become endemic in the population is lowered. The endemic equilibrium solution is also affected, with an increased proportion of the population in the treatment class and a decreased proportion in the susceptible class. Including totally recovery does not determine whether the proportion of individuals with alcohol problems increases or decreases, however it does effect the size of the change. Parameter estimates are made from information regarding binge drinking where we find an increase in the recovery rate decreases the proportion of binge drinkers in the population.     

Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel &; Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser &; Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.     

Intrinsic contribution of the 2'-hydroxyl to RNA conformational heterogeneity

Canonical duplex RNA assumes only the A-form conformation at the secondary structure level while, in contrast, a wide range of noncanonical, tertiary conformations of RNA occur. Here, we show how the 2'-hydroxyl controls RNA conformational properties. Quantum mechanical calculations reveal that the orientation of the 2'-hydroxyl significantly alters the intrinsic flexibility of the phosphodiester backbone, favoring the A-form in duplex RNA when it is in the base orientation and facilitating sampling of a wide range of noncanonical, tertiary structures when it is in the O3' orientation. Influencing the orientation of the 2'-hydroxyl are interactions with the environment, as evidenced by crystallographic survey data, indicating the 2'-hydroxyl to sample more of the O3' orientation in noncanonical RNA structures. These results indicate that the 2'-hydroxyl acts as a "switch", both limiting the conformation of RNA to the A-form at the secondary structure level and allowing RNA to sample a wide range of noncanonical tertiary conformations.     

Reverse cope elimination of hydroxylamines and alkenes or alkynes: theoretical investigation of tether length and substituent effects

Quantum mechanical calculations have been used to study the intramolecular additions of hydroxylamines to alkenes and alkynes ("reverse Cope eliminations"). In intermolecular reverse Cope eliminations, alkynes are more reactive than alkenes. However, competition experiments have shown that tethering the hydroxylamine to the alkene or alkyne can reverse the reactivity order from that normally observed. The exact outcome depends on the length of the tether. In agreement with experiment, a range of density functional theory methods and CBS-QB3 calculations predict that the activation energies for intramolecular reverse Cope eliminations follow the order 6-exo-dig < 5-exo-trig < 5-exo-dig ≈ 7-exo-dig. The order of the barriers for the 5-, 6-, and 7-exo-dig reactions of alkynes arises mainly from differences in tether strain in the transition states (TSs), but is also influenced by the TS interaction between the hydroxylamine and alkyne. Cyclization onto an alkene in the 5-exo-trig fashion incurs slightly less tether strain than a 6-exo-dig alkyne cyclization, but its activation energy is higher because the hydroxylamine fragment must distort more before the TS is reached. If the alkene terminus is substituted with two methyl groups, the barrier becomes so much higher that it is also disfavored compared to the 5- and 7-exo-dig cyclizations.     

A distal pocket Leu residue inhibits the binding of O2 and NO at the distal heme site of cytochrome c'

Cytochromes c' are pentacoordinate heme proteins with sterically hindered distal sites that bind NO and CO but do not form stable complexes with O(2). Removal of distal pocket steric hindrance via a Leu→Ala mutation yields favorable O(2) binding (K(d) ~49 nM) without apparent H-bond stabilization of the Fe-O(2) moiety, as well as an extremely high distal heme-NO affinity (K(d) ~70 fM). The native Leu residue inhibits distal coordination of diatomic ligands by decreasing k(on) as well as increasing k(off). The connection between distal steric constraints, k(off) values, and distal to proximal heme-NO conversion is discussed.     

Heme flattening is sufficient for signal transduction in the H-NOX family

The H-NOX family of nitric oxide (NO) sensing proteins has received considerable attention because its members include the mammalian NO sensor, soluble guanylate cyclase. Despite this attention, the mechanism of signal transduction has not been elucidated. Structural studies of bacterial members of the family have revealed that the H-NOX heme cofactor is extremely distorted from planarity. Furthermore, it has been determined that heme distortion is maintained primarily by a conserved proline residue located in the proximal heme pocket. It has been suggested that changes in heme planarity may contribute to signal transduction. Here we demonstrate that heme flattening is, indeed, sufficient for signal transduction in the H-NOX family. Using our previously described H-NOX/diguanylate cyclase functional partners from Shewanella woodyi, we demonstrate that mutation of the conserved proline (P117 in SwH-NOX) to alanine, which results in heme flattening, has the same affect on phosphodiesterase activity as NO binding to wildtype SwH-NOX. This study demonstrates, for the first time, that heme flattening mimics the activated, NO-bound state of H-NOX and suggests that NO binding induces heme flattening as part of the signal transduction mechanism in the H-NOX family.