Those training to become teachers in England during the 2019/20 academic year were severely impacted by first national lockdown due Covid-19 pandemic, with many missing school placements, giving them less time build experience and confidence before becoming newly qualified (NQTs). Their of teaching was also pandemic. As part a British Academy-funded project, we collected data from 2020/1 NQTs t...

We investigate whether and how personal mathematical knowledge at an advanced level impacts on teaching at a lower school level. We study this in the context of functions because understanding them permeates secondary and advancedmathematics. Textbook treatment of these can be patchy, implying a need for knowledgeable teachers to rectify weaknesses. A small observational study of two mathematic...

For a prime power q > 1 and an integer n ≥ 2 we consider the extension Fqn/Fq of Galois fields together with a cyclic Fq-vector space endomorphism τ . An element of Fqn is called a primitive τ-generator over Fq provided it generates the multiplicative group of Fqn , as well as the additive group of Fqn regarded as an Fq[x]-module with respect to τ . The notion of a primitive τ -generator genera...

We will examine the subset FQ,p of Farey fractions of order Q consisting of those fractions whose denominators are not divisible by a fixed prime p. In particular, we will provide an asymptotic result on the distribution of H−tuples of consecutive fractions in FQ,p, as Q → ∞.

Let Fq be a finite field and consider an extension Fqn where an optimal normal element exists. Using the trace of an optimal normal element in Fqn , we provide low complexity normal elements in Fqm , with m = n/k. We give theorems for Type I and Type II optimal normal elements. When Type I normal elements are used with m = n/2, m odd and q even, our construction gives Type II optimal normal ele...

Let Fv and Fe be sets of faulty vertices and faulty edges, respectively, in the folded hypercube FQn so that |Fv| + |Fe| ≤ n − 2, for n ≥ 2. Choose any fault-free edge e. If n ≥ 3 then there is a fault-free cycle of length l in FQn containing e, for every even l ranging from 4 to 2 − 2|Fv |; if n ≥ 2 is even then there is a fault-free cycle of length l in FQn containing e, for every odd l rangi...

This paper presents an explicit construction for an ((n = 2qt, k = 2q(t−1), d = n−(q+1)), (α = q(2q)t−1, β = α q )) regenerating code over a field FQ operating at the Minimum Storage Regeneration (MSR) point. The MSR code can be constructed to have rate k/n as close to 1 as desired, sub-packetization level α ≤ r nr for r = (n− k), field size Q no larger than n and where all code symbols can be ...