Derek Holton

Scaffolding and Metacognition

This paper proposes an expanded conception of scaffolding with four key elements: i. scaffolding agency – expert, reciprocal, and self-scaffolding; ii. scaffolding domain – conceptual and heuristic scaffolding; iii. the identification of self-scaffolding with metacognition; and iv. the identification of six zones of scaffolding activity; each zone distinguished by the matter under construction and the relative positioning of the participant(s) in the act of scaffolding. These key elements are illustrated with empirical examples drawn from a variety of research studies. Scaffolding, thus conceived, brings together several theoretical domains, and by situating metacognition within a framework derived from the social activity of scaffolding, a bridge is formed between the instructional support a teacher might provide and the learners self-control of the learning process. With regard to instruction and the role of the teacher, it is the authors’ contention that a major object of instruction is the progressive relocation of scaffolding agency in the direction of the learner with the long-term goal of equipping the learner to take control of their own learning.

The smallest non-Hamiltonian 3-connected cubic planar graphs have 38 vertices

Abstract We show that all 3-connected cubic planar graphs on 36 or fewer vertices are hamiltonian, thus extending results of Lederberg, Butler, Goodey, Wegner, Okamura, and Barnette. Furthermore, the only non-hamiltonian examples on 38 vertices which are not cyclically 4-connected are the six graphs which have been found by Lederberg, Barnette, and Bosak.

Technology as a Tool for Teaching Undergraduate Mathematics

In this chapter we review the current situation regarding the use of technology as a teaching tool at university level When we talk about technology we mean graphic calculators and CD-ROM as well as all aspects of computers including software and their use with the Internet. However, we have chosen to emphasize here the theoretical perspectives and content areas of undergraduate research rather than the tools through which these are implemented.

Algorithms for Pattern Involvement in Permutations

We consider the problem of developing algorithms for the recognition of a fixed pattern within a permutation. These methods are based upon using a carefully chosen chain or tree of subpatterns to build up the entire pattern. Generally, large improvements over brute force search can be obtained. Even using on-line versions of these methods allow for such improvements, though often not as great as for the full method. Furthermore, by using carefully chosen data structures to fine tune the methods, we establish that any pattern of length 4 can be detected in O(n log n) time. We also improve the complexity bound for detection of a separable pattern from O(n6) to O(n5 log n).

Nonhamiltonian 3-Connected Cubic Planar Graphs

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp, and we present all nonhamiltonian examples of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. The fact that all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified.

Removable edges in 3-connected graphs

An edge e of a 3-connected graph G is said to be removable if G - e is a subdivision of a 3-connected graph. If e is not removable, then e is said to be nonremovable. In this paper, we study the distribution of removable edges in 3-connected graphs and prove that a 3-connected graph of order n ≥ 5 has at most [(4 n — 5)/3] nonremovable edges.

On the 2-extendability of planar graphs

Abstract Some sufficient conditions for the 2-extendability of k -connected k -regular ( k ⩾3) planar graphs are given. In particular, it is proved that for k ⩾3, a k -connected k -regular planar graph with each cyclic cutset of sufficiently large size is 2-extendable.

Peer tutoring in first-year undergraduate mathematics

A peer tutoring approach was taken for part of the teaching of mathematics to two different classes at a tertiary institution, most of whose students were preparing to be teachers at either the primary or secondary levels. It was hoped that peer tutoring would increase the learning and understanding of the students involved. As many of the students are training to become teachers, it was hoped that the work on peer tutoring might have particular relevance for them. From qualitative evidence, the experiment appears to have been successful. The vast majority of the students would like the experience to be repeated during their mathematics course.

Permutations of a Multiset Avoiding Permutations of Length 3

Abstract We consider permutations of a multiset which do not contain certain ordered patterns of length 3. For each possible set of patterns we provide a structural description of the permutations avoiding those patterns, and in many cases a complete enumeration of such permutations according to the underlying multiset.

Lower bound of cyclic edge connectivity for n -extendability of regular graphs

Abstract A cyclically m-edge-connected n-connected k-regular graph is called an (m.n.k) graph. It is proved that for any m > 0 and k ⩾3, there is an (m, k, k) bipartite graph. A graph G is n-extendable if every matching of size n in G lies in a perfect matching of G. We prove the existence of a (k2 −1, k + 1, k + 1) bipartite graph which is not k-extendable and the existence of an (m, k + 1, k + 1) graph which is not n-extendable, where n ⩾ 2, k ⩾ 2 and m is any positive integer. The existence of the former graphs shows that a result of Holton and Plummer is sharp.