Deborah King

The impact of assumed knowledge entry standards on undergraduate mathematics teaching in Australia

Over the last two decades, many Australian universities have relaxed their selection requirements for mathematics-dependent degrees, shifting from hard prerequisites to assumed knowledge standards which provide students with an indication of the prior learning that is expected. This has been regarded by some as a positive move, since students who may be returning to study, or who are changing career paths but do not have particular prerequisite study, now have more flexible pathways. However, there is mounting evidence to indicate that there are also significant negative impacts associated with assumed knowledge approaches, with large numbers of students enrolling in degrees without the stated assumed knowledge. For students, there are negative impacts on pass rates and retention rates and limitations to pathways within particular degrees. For institutions, the necessity to offer additional mathematics subjects at a lower level than normal and more support services for under-prepared students impacts on workloads and resources. In this paper, we discuss early research from the First Year in Maths project, which begins to shed light on the realities of a system that may in fact be too flexible.

Linear and cyclic distance-three labellings of trees

Given a finite or infinite graph GG and positive integers l,h1,h2,h3l,h1,h2,h3, an L(h1,h2,h3)L(h1,h2,h3)-labelling of GG with span ll is a mapping f:V(G)→{0,1,2,…,l}f:V(G)→{0,1,2,…,l} such that, for i=1,2,3i=1,2,3 and any u,v∈V(G)u,v∈V(G) at distance ii in GG, |f(u)−f(v)|≥hi|f(u)−f(v)|≥hi. A C(h1,h2,h3)C(h1,h2,h3)-labelling of GG with span ll is defined similarly by requiring |f(u)−f(v)|l≥hi|f(u)−f(v)|l≥hi instead, where |x|l=min{|x|,l−|x|}|x|l=min{|x|,l−|x|}. The minimum span of an L(h1,h2,h3)L(h1,h2,h3)-labelling, or a C(h1,h2,h3)C(h1,h2,h3)-labelling, of GG is denoted by λh1,h2,h3(G)λh1,h2,h3(G), or σh1,h2,h3(G)σh1,h2,h3(G), respectively. Two related invariants, λh1,h2,h3∗(G) and σh1,h2,h3∗(G), are defined similarly by requiring further that for every vertex uu there exists an interval Iumod(l+1) or modl, respectively, such that the neighbours of uu are assigned labels from IuIu and Iv∩Iw=0Iv∩Iw=0 for every edge vwvw of GG. A recent result asserts that the L(2,1,1)L(2,1,1)-labelling problem is NP-complete even for the class of trees. In this paper we study the L(h,p,p)L(h,p,p) and C(h,p,p)C(h,p,p) labelling problems for finite or infinite trees TT with finite maximum degree, where h≥p≥1h≥p≥1 are integers. We give sharp bounds on λh,p,p(T)λh,p,p(T), λh,p,p∗(T), σh,1,1(T)σh,1,1(T) and σh,1,1∗(T), together with linear time approximation algorithms for the L(h,p,p)L(h,p,p)-labelling and the C(h,1,1)C(h,1,1)-labelling problems for finite trees. We obtain the precise values of these four invariants for complete mm-ary trees with height at least 4, the infinite complete mm-ary tree, and the infinite (m+1)(m+1)-regular tree and its finite subtrees induced by vertices up to a given level. We give sharp bounds on σh,p,p(T)σh,p,p(T) and σh,p,p∗(T) for trees with maximum degree Δ≤h/pΔ≤h/p, and as a special case we obtain that σh,1,1(T)=σh,1,1∗(T)=2h+Δ−1 for any tree TT with Δ≤hΔ≤h.

Maximal entropy of permutations of even order

A finite invariant set of a continuous map of an interval induces a permutation called its type. If this permutation is a cycle, it is called its orbit type. It has been shown by Geller and Tolosa that Misiurewicz–Nitecki orbit types of period

A Lower Bound for the Maximum Topological Entropy of

n

(4k+2)

congruent to

-Cycles

1

Maximum Entropy of Cycles of Even Period.

(mod 4) and their generalizations to orbit types of period

Investigating students’ perceptions of graduate learning outcomes in mathematics

n

Classification of permutations and cycles of maximum topological entropy

congruent to

Building a Network and Finding a Community of Practice for Undergraduate Mathematics Lecturers

3