Enhancing Small Group Teaching in Plant Sciences: A Research and Development Project in Higher Education

The Department of Plant Sciences at the University of Cambridge uses a range of learning and teaching environments including lectures, practical laboratories and small group tutorials'. Under the auspices of the Cambridge-MIT Institute's Pedagogy Programme, a two-year research and development project concerned with the development of small-group teaching is being undertaken. The research element of this project endeavours to illuminate current practice and identify areas in which evidence-based development might take place. The development element will include professional development activities and the production of curriculum resources including appropriate online material. This is a multi-method study including a series of student questionnaires; focus groups of students; semi-structured interviews with staff members; and the collection of video of small group teaching. In this paper we report selected findings from the 'student data' of the first year of this project.The questionnaire, conducted with two cohorts of students (2nd and 3rd year Undergraduates), used a double-scale questionnaire in which students were asked to report both on the prevalence of a range of teaching and learning practices and on how valuable these were in supporting their learning. This type of questionnaire instrument is particularly appropriate because the data it generates is suggestive of areas for changes in practice. The gaps between 'practices' and 'values' (across both cohorts) suggested that students valued activities which improved their understanding of how elements of the course were interrelated; which related course content to 'authentic' examples; and those in which teachers made explicit the characteristics of 'high quality' student work. Small group teaching, in the view of most students, was best used to extend and explore concepts introduced in lectures rather than simply reinforcing them or assessing student understanding.Data gathered through focus group activities illuminated the questionnaire data, providing detailed accounts of how students managed their own learning, and the roles played in this by lectures, small group teaching and other resources. Students identified the processes of planning and writing essays as key learning activities during which they integrated diverse course content and reflected on problematic knowledge. Questionnaire and focus group data suggested that students had less clear views regarding the value of collaborative learning, peer-assessment or activities such as making presentations to other students. When students talked in positive terms about these activities, they often referred to the learning benefits of preparation for the tasks rather than of the collaborative activities themselves. These views may provide indications of potential barriers to changes in learning and teaching environments, and suggest that any such changes may have to be carefully justified to students in terms of benefits to their own learning. Many of our findings are broadly in accord with other work on teaching and learning in Higher Education settings (such as the 'Oxford Learning Context Project' and the 'Enhancing Teaching-Learning Environments in Undergraduate Courses' Project) in that 'deep learning' and 'authenticity' in learning activities are valued by students, and that the introduction of specific formative practices (such as sharing notions of 'quality') would be welcomed. At the same time, amongst the students in our sample, a view of learning as an individual process of 'learning-as-acquisition' predominates over a view that it is a social process of 'learning-as-participation', and this will inform the planning of the 'development' aspect of the project. We conclude with a discussion of how the approach we have used might be more widely applied both within and beyond the Cambridge-MIT partnership. We also identify potential affordances of, and barriers to, the development of research-informed teaching in Higher Education

Current moments of 1D ASEP by duality

We consider the exponential moments of integrated currents of 1D asymmetric simple exclusion process using the duality found by Sch\"utz. For the ASEP on the infinite lattice we show that the $n$th moment is reduced to the problem of the ASEP with less than or equal to $n$ particles.Comment: 13 pages, no figur

Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]Comment: 16 pages, LaTeX file, no figures. Revision has minor change

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations, added reference [19], fit into margins, performed minor editin

Random walks and random fixed-point free involutions

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page

{\bf τ\tau-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $\tau$-functions for certain Painlev\'e systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $\tau$-functions for certain Painlev\'e systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $\tau$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $\tau$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $\tau$-functions is the gap probability in the corresponding orthogonal symmetry case.Comment: AMS-Late

Effect of spin-orbit coupling on the excitation spectrum of Andreev billiards

We consider the effect of spin-orbit coupling on the low energy excitation spectrum of an Andreev billiard (a quantum dot weakly coupled to a superconductor), using a dynamical numerical model (the spin Andreev map). Three effects of spin-orbit coupling are obtained in our simulations: In zero magnetic field: (1) the narrowing of the distribution of the excitation gap; (2) the appearance of oscillations in the average density of states. In strong magnetic field: (3) the appearance of a peak in the average density of states at zero energy. All three effects have been predicted by random-matrix theory.Comment: 5 pages, 4 figure

Magnesium-uranium alloy system

Analytical, X-ray, thermal, and metallographic data have been obtained in the study of magnesium-uranium system, and a proposed phase diagram has been constructed

Crossover to the KPZ equation

We characterize the crossover regime to the KPZ equation for a class of one-dimensional weakly asymmetric exclusion processes. The crossover depends on the strength asymmetry $an^{2-\gamma}$ ($a,\gamma>0$) and it occurs at $\gamma=1/2$. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if $\gamma\in(1/2,1]$, while for $\gamma=1/2$ it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained.Comment: Published by Annales Henri Poincare Volume 13, Number 4 (2012), 813-82

Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. We conclude with a brief discussion on the derivation of the probability distribution function of the corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and Gaussian Symplectic Ensembles (GSEn)