Abstract:
Any curriculum is a construct of perceived social, political and economic needs developed at a point in time. Given that these needs are in a constant state of flux, the curriculum is subjected to periodical renewal and development processes. Gaining more visibility in the iterations of curriculum documentation is the need for Australians to be more aware of their activities impacting on the environment. Comparable to a specific curriculum document, the content knowledge delivered through initial teacher education is specific to the conditions at a point in time, requiring teachers to adapt as the curriculum evolves. Peering through the lens of teacher content knowledge, research has shown that teachers need to efficiently adapt to these changes and effectively develop their expertise in the new content material. Those that can innovate in applying their existing knowledge to the new content are said to possess adaptive expertise. Given the breadth and diversity of school curriculum, the economisation of formalised professional learning opportunities does not address the shortfall in teacher content knowledge. As a result, qualified teachers have resorted to autonomous methods of professional learning to bridge the knowledge gap. This study examines whether autonomous professional learning approaches are an effective method for teachers to gain an understanding of new syllabus content. Using a case study of technology education teachers self-educating around the concepts of eco-friendly technology education, the study identifies the intrinsic motivation of teachers to know and understand their evolving subject, and provides a basis for self-directed and autonomous professional learning. What this results in is the successful development of a basic understanding of new information and concepts in technology education.

Abstract:
The popularity of sites like YouTube demonstrates the potential
preference for users to use video podcasts (vodcasts) as an instructional tool.
As educators have been encouraged to become more literate in authoring Web 2.0
technologies, the implementation of vodcasts as an effective pedagogy has been
increasingly used in educational settings. The recent proliferation of distance
education courses caused educators in higher education settings to consider why
some programs are more suitable for distance education than others, and whether
procedural knowledge and skills that are required in various industries may be
able to be delivered using contemporary technologies. This article discusses
the efficacy of using vodcasting as a pedagogical tool, in developing procedural
knowledge and skills in computer aided design and drawing, to pre-service
teachers studying via distance education, and demonstrates the capacity for
vodcasts to foster autonomous student learning.

Abstract:
The development of universities as participants in the cultural life of a city is a new and welcome development that in many cases has been led by the university’s museums. This paper discusses the resulting change in the perception of university museums, both within the institution and among the broader community, and argues that university museums have a unique potential to be cultural players and leaders within their university and community. With direct call on experts across a diverse range of subject areas they have an ability to provide an extraordinary range of public programs from performances, concerts, and artist interventions to more traditional lectures and forums. The benefits can be enticing. For some museums that have gone down this path and strategically aligned themselves with their university’s goals, these developments have lead to an increased visibility (internal and external) and funding (including new buildings and increased staffing numbers). Case study: The University of Sydney’s Museums and their role in developing and defining the university as a cultural precinct.

Abstract:
We call a family G of subsets of [n] a k-generator of (\mathbb{P}[n]) if every (x \subset [n]) can be expressed as a union of at most k disjoint sets in (\mathcal{G}). Frein, Leveque and Sebo conjectured that for any (n \geq k), such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl \cite{alon} in order to show that for fixed k, any k-generator of (\mathbb{P}[n]) must have size at least (k2^{n/k}(1-o(1))), thereby verifying the conjecture asymptotically for multiples of k.

Abstract:
A family of permutations (\mathcal{A} \subset S_{n}) is said to be (t)-\textit{intersecting} if any two permutations in (\mathcal{A}) agree on at least (t) points, i.e. for any (\sigma, \pi \in \mathcal{A}), (|\{i \in [n]: \sigma(i)=\pi(i)\}| \geq t). It was recently proved by Friedgut, Pilpel and the author that for (n) sufficiently large depending on (t), a (t)-intersecting family (\mathcal{A} \subset S_{n}) has size at most ((n-t)!), with equality only if (\mathcal{A}) is a coset of the stabilizer of (t) points (or `(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first obtain a rough stability result for (t)-intersecting families of permutations, namely that for any (t \in \mathbb{N}) and any positive constant (c), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family of permutations of size at least (c(n-t)!), then there exists a (t)-coset containing all but at most a (O(1/n))-fraction of (\mathcal{A}). We use this to prove an exact stability result: for (n) sufficiently large depending on (t), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family which is not contained within a (t)-coset, then (\mathcal{A}) is at most as large as the family \mathcal{D} & = & \{\sigma \in S_{n}: \sigma(i)=i \forall i \leq t, \sigma(j)=j \textrm{for some} j > t+1\} && \cup \{(1 t+1),(2 t+1),...,(t t+1)\} which has size ((1-1/e+o(1))(n-t)!). Moreover, if (\mathcal{A}) is the same size as (\mathcal{D}) then it must be a `double translate' of (\mathcal{D}), meaning that there exist (\pi,\tau \in S_{n}) such that (\mathcal{A}=\pi \mathcal{D} \tau). We also obtain an analogous result for (t)-intersecting families in the alternating group (A_{n}).

Abstract:
A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {\sigma \in S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \geq 4, if A,B \subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author.

Abstract:
We consider the problem of finding the maximum possible size of a family of k-dimensional subcubes of the n-cube {0,1}^{n}, none of which is contained in the union of the others. (We call such a family `irredundant'). Aharoni and Holzman conjectured that for k > n/2, the answer is {n choose k} (which is attained by the family of all k-subcubes containing a fixed point). We give a new proof of a general upper bound of Meshulam, and we prove that for k >= n/2, any irredundant family in which all the subcubes go through either (0,0,...,0) or (1,1,...,1) has size at most {n choose k}. We then give a general lower bound, showing that Meshulam's upper bound is always tight up to a factor of at most e.

Abstract:
We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of 2 points. We also obtain a Hilton-Milner type result, namely that if $\mathcal{A}$ is such a family which is not contained within a coset of the stabilizer of 2 points, then it is no larger than the family $\{\sigma \in S_{n}:\ \sigma(1)=1,\sigma(2)=2,\ \#\{\textrm{fixed points of}\sigma \geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\ 4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}$. We conjecture that for $t \in \mathbb{N}$, and for $n$ sufficiently large depending on $t$, if $\mathcal{A}$ is family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly $t-1$ times, then $|\mathcal{A}| \leq (n-t)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of $t$ points. This can be seen as a permutation analogue of a conjecture of Erd\H{o}s on families of $k$-element sets with a forbidden intersection, proved by Frankl and F\"uredi in [P. Frankl and Z. F\"uredi, Forbidding Just One Intersection, Journal of Combinatorial Theory, Series A, Volume 39 (1985), pp. 160-176].

Abstract:
We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $\delta |A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| \delta \log_{2}(1/\delta) = 2^{t}(n-t+\delta \log_{2}(1/\delta))$, provided $\delta$ is less than an absolute constant. This is sharp whenever $\delta = 1/2^{j}$ for some $j \in \{1,2,\ldots,t\}$.

Abstract:
A family of permutations A \subset S_n is said to be t-set-intersecting if for any two permutations \sigma, and \pi \in A, there exists a t-set x whose image is the same under both permutations, i.e. \sigma(x)=\pi(x). We prove that if n is sufficiently large depending on t, the largest t-set-intersecting families of permutations in S_n are cosets of stabilizers of t-sets. The t=2 case of this was conjectured by J\'anos K\"orner. It can be seen as a variant of the Deza-Frankl conjecture, proved in [4]. Our proof uses similar techniques to those of [4], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.